Wikiversity:Colloquium

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&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

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

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

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

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

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

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

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

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

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

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

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

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

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

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

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)</text>
      <sha1>lbt9x0bks922j8gzsxrm2vw8pa09zk0</sha1>
    </revision>
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  <page>
    <title>Wikiversity:Copyrights</title>
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        <username>Kwamikagami</username>
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      <text bytes="3033" sha1="tqclvvgpp0bgqnbf6cxihaujfdte19f" xml:space="preserve">{{proposal}}

[[Image:Unicode 1xF12F.svg|thumb|upright|Copyright/copyleft Logo]]
{{TOCright}}
The license [[Wikiversity]] uses grants free access to our [[learning resource|content]] in the same sense as [[w:free software|free software]] is licensed freely. This principle is known as '''[[w:copyleft|copyleft]]'''. That is to say, Wikiversity content can be copied, modified, and redistributed, either commercially or noncommercially, ''so long as'' the new version grants the same freedoms to others and acknowledges the authors of the Wikiversity content used (a direct link back to the article satisfies our author credit requirement).  Wikiversity [[learning resource]]s therefore will remain free forever and can be used by anybody subject to certain restrictions, most of which serve to ensure that freedom.

To fulfill the above goals, the text contained in Wikiversity is licensed to the public under the [http://creativecommons.org/licenses/by-sa/4.0/ Creative Commons Attribution/Share-Alike License] and '''[[Wikiversity:GNU Free Documentation License|GNU Free Documentation License]]''' (GFDL). Please see [http://wikimediafoundation.org/wiki/Terms_of_Use Terms of Use].

For material whose copyright is in question see [[Wikiversity:Copyright issues]]. This also includes information on how to deal with such material.

==Copyright and Media Uploads==
According to [[foundation:Resolution:Licensing_policy|Foundation policy]], uploads are required to have a [[Wikiversity:License tags|license template]] and source. To add this information after upload, please edit the image description page directly (click the ''edit this page'' tab); uploading a new version of the media ''will not'' update the license template, nor the source information.

== See also ==

===Wikiversity===
* [[Wikiversity:License tags]]
* [[Wikiversity:Uploading files]]
* [[foundation:Resolution:Licensing_policy|Foundation licensing policy]]
* [[The GFDL and you]]

===Wikipedia===
* [[w:Wikipedia:Copyright FAQ|Wikipedia:Copyright FAQ]]
* The [[w:Wikipedia:Contributing FAQ|Wikipedia:Contributing FAQ]] for questions on copyright.
* [[w:Wikipedia:Mirrors and forks|Wikipedia:Mirrors and forks]]
* [[w:Wikipedia:Standard license violation letter|Wikipedia:Standard license violation letter]]
* [[w:Wikipedia:Copyright problems|Wikipedia:Copyright problems]]
* [[w:Wikipedia:Spotting possible copyright violations|Wikipedia:Spotting possible copyright violations]]
* [[w:Wikipedia:Fair use|Wikipedia:Fair use]]
* [[w:Wikipedia:Image copyright tags|Wikipedia:Image copyright tags]]
* [[w:Wikipedia:Boilerplate request for permission|Wikipedia:Boilerplate request for permission]]
* [[w:Wikipedia:Copyright issues|Wikipedia:Copyright issues]]

===Meta===
* [[m:Wikipedia and copyright issues|Wikipedia and copyright issues]]
* [[m:Avoid Copyright Paranoia|Avoid Copyright Paranoia]]
* [[m:Permission grant extent|Permission grant extent]]
* [[w:GSFDL|A new GFDL]]?

{{official policies}}
{{proposed policies}}

[[Category:Wikiversity copyrights]]</text>
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  <page>
    <title>Bloom Clock</title>
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[[File:Sommerblumen01.JPG‎|thumb|right|300px|alt=A field of flowers|Summertime in Germany, 1975]]
{{TOCright}}

The '''Bloom Clock''' is a research and learning project about flowering plants. The research component is aimed at creating a language for discussing the bloom times of wildflowers and other plants that is neutral in respect to climate, region, and hemisphere, while the learning project aspect is aimed at helping people identify plants using visual keys.

To participate, read the page on '''[[/How to Contribute/]]''', then go to the [[/Contributors/]] page and sign in to your account, and then start adding your records in either of two ways:
#Participants who know plants fairly well (including the scientific names) can log bloom sightings on [[Bloom Clock/Current|the current text-based clock]] or using the [[/Master List/]].
#Participants can also use the '''[[/Keys|keys]]''', which can help you narrow down and identify plants through simple characteristics and photographic images. 

If you want to use the clock to identify a flower you saw today, try using the '''[[/Keys/]]'''!

If you are interested in improving the clock or have questions and comments about it, please join us on the [[/Project Discussion/]] page.

==What are Bloom Clocks?==

Bloom clocks are kept by gardeners, ecologists, and others to record the time of year different plants are in bloom.

==How are bloom clocks helpful?==
The data from bloom clocks tell us about both the plants themselves, and the region in which a particular plant is growing. 
*Knowing when a plant blooms (relative to other plants) is helpful for garden designers.
*Knowing when nectar-producing plants bloom is useful for farmers, orchardists, and beekeepers who want to ensure a continual supply of nectar.
*Knowing when wind-pollinated plants bloom can help those with allergies (and the doctors that treat them) predict when pollen will be a problem. A major example of this is that in most parts of North America, goldenrods (showy, yellow flowers that don't cause allergies) tend to bloom at the same time as ragweeds (which have green, non-showy flowers, and ''do'' cause allergies). 
*In [[w:Integrated Pest Management|Integrated Pest Management]] (a strategy used by farmers and gardeners in pest and disease control), bloom clocks can provide [[w:Phenology|phenological cues]] which tell the farmer or gardener when to look out for a certain pest. For example, if a certain pest generally emerges at the same time as a particular plant is blooming, the farmer or gardener will know to check susceptible crops and plants for signs of the pest.
*Patterns in the variation of bloom times with temperature or day length can reveal aspects of plant physiology and growth season relevant to modeling plant response to the environment.  Such models help reveal seasonal roles plants play in microclimates and in cycles of nutrients and water.
*Bloom times can be used as indicators for monitoring changes of local and regional climate. Flowers have been blooming earlier in the 21st century, indicating accelerating climate change.&lt;ref&gt;{{cite web |title=Flowers arriving a month early in UK as climate heats up |url=https://www.theguardian.com/environment/2022/feb/02/flowers-arriving-month-early-uk-climate-heats-up-bloom-insects-birds |work={{w|The Guardian}} |author=Damian Carrington |date=2022-02-02 |accessdate=2022-02-02}}&lt;/ref&gt;
===How can the Wikiversity Bloom Clock help support other Wikimedia projects?===
:''When the Bloom Clock project was developed, it was hoped that original research could support other Wikimedia projects. It is possible, but not in this format. See [[WikiJournal]] for the accepted approach necessary to support other Wikimedia projects. The following comments represent the goals of this project when it was developed.''
 
*For the Wikipedias and garden books on the Wikibooks, the bloom clock data can eventually yield a &quot;geographically neutral&quot; language for discussing the flowering times of plants, which will be helpful in writing articles.
*For Wikimedia Commons, the visual interface pages can help uploaders identify plants (by using DynamicPageLists for flowers of certain colors and seasons).

==How can a bloom clock be created on Wikiversity?==

If there are enough participants, all it should require is for bloom watchers to note any plants they noticed to be in bloom on a particular day, with information on their location specific enough to determine the local [[w:growing degree day|growing degree days]] and day length, which are the two factors plants respond to. Precipitation reports may also be handy for some climates.

Each report can then be organized to reduce the effects of anomalous data, allowing us to generate geographical &quot;zones&quot; that can eventually be used when describing a plant's expected bloom time in a particular region. 

Data can be extrapolated any number of times. Many different species can be used, because they can be correlated to the bloom times of other species over time.

==System==

===Hypothesis===
*Plants may be classified into groups that can accurately predict bloom dates according to geographical zones.
*There will be 2 sorts of zones for most plants: a day-length zone and a growing-degree-day zone (gdd-zone).
*Some plants will not fit into this system.

===Plant behaviors===
* Some plants bloom according to growing degree days.
* Some plants bloom according to day length.
* Some plants bloom according to various conditions.
* The gene Apetala1 controls the processes responsible for blooming based on environmental cues. &lt;ref&gt;{{citation|url=http://www.livescience.com/32529-how-do-flowers-know-when-to-bloom.html|title=
How Do Flowers Know When to Bloom?}}&lt;/ref&gt;

===Methods===
*Collect data.
*Collate data.
*Establish zones.

===Further testing===
*Check plant bloom times in areas not previously checked, but in the same zones.

==References==
{{reflist}}

[[Category:Bloom Clock| ]]
[[Category:Biology learning projects]]
[[Category:Primary research proposals]]
[[Category:Research projects]]</text>
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  <page>
    <title>Introduction to Computers/Networks</title>
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      <text bytes="20281" sha1="e1n7yumkeimjmbdvv20vnfuj51z6szr" xml:space="preserve">&lt;big&gt;Course Navigation&lt;/big&gt;
{| class=&quot;wikitable&quot;
! [[Introduction_to_Computers/Personal|'''&lt;&lt;  Previous - Personal''']]
!                                                        
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! [[Introduction_to_Computers/Security|'''Next - Security  &gt;&gt; ''']]
|}

{{Introduction to computers/header}}

[[Image:Internet map 1024.jpg|thumb|300px|Visualization of the various routes through a portion of the Internet.]]
[[Image:Sample-network-diagram.png|thumb|right|400px|A sample network diagram]]
A network is a group of computers (or a group of smaller networks) that are connected to each other by various means, so that they may communicate with each other. The [[Introduction to Computers/Internet|internet]] is the largest network in the world.

==Benefits==
==Resource Sharing==
[[Image:networkprinter.jpg|thumb|200px|Network Printer]]

Networks are able to share one resource, such as a printer, for numerous computers.  This allows many individual computers to access a single network resource.  This saves money and space for the organization. [[w:Computer_printer]]

===Data===
[[Image:eb470f1.jpg.jpg|thumb|200px|Data]]
Once networks were setup, people found that the next best thing was the ability to easily share documents. The type of server that stores files is called a file server.
Data can be in the form of text, images, numbers or characters. [[w:Data]]

===Programs===
[[Image:network.jpg|thumb|right|100px|Network based programs]]
Server or Network based programs are programs that are loaded onto an online server or network as opposed to directly onto the individual computers. This program can therefor be accessed by any computer possessing the correct amount of bandwidth and system specifications. In short it uses the server as an application server.

===Work better===

Many organizations use networks for many purposes such as making schedule for colleagues, pick up days for meetings (when everybody will be able to attend) and provide useful online connections for network-linked employees.

Employees can communicate on a network with other employees through email.

[[w:Computer_networking]]

==Types==
===mainframe===
Mainframe computers are typically large, metal boxed computers with large processing abilities. The terminals are called &quot;dumb terminals&quot; because they only send and receive data, leaving the processing to the mainframe.  [[w:Mainframe_computer]]

===Client server===
The client/server refers to the way two computer programs interact with each other. The client makes a request from the server, who then fulfills the request. Although this idea can be used on one computer it is an efficient way for a network of computers in different locations to interconnect. [http://searchnetworking.techtarget.com/sDefinition/0,,sid7_gci211796,00.html]
====LAN====

LAN stands for Local Area Network.  The first LANs were created in the late 1970s.
LANs are small networks constricted to a small area like a house, office, or city.
LANs are used to share resources like storage,internet,etc...
[[w:Local_area_network]]

A 'node' on a LAN is a connected computer or device like a printer.

====WAN====

WAN stands for Wide Area Network.  WANs are very large networks that interconnect smaller LAN networks,
for a large geographic area like a country(i.e., any network whose communications links cross metropolitan, regional, or national boundaries.) [[w:Wide_area_network]]

WANs are usually for private companies, however, some built by internet service providers connect LANs to the internet.

*WAN can use a combination of satellites, microwave, and link and variety of computers from mainframes to terminals.

A 'node' on a WAN is a LAN.

====MAN====

When the LANs that you want to connect are not far apart, just blocks away, then you can make a MAN (Metropolitan Area Network).  The main difference between a WAN and a MANs is the speed of the connection. Because the LANs are so close in a MAN, high speed fiber optic cables are affordable.
[[w:Metropolitan_area_network]]

====PAN====
Personal Area Network: a network on you. Usually, uses short range wireless technology is used to connect devises like a cell phone and a PDA. [[w:Personal area network]]

====HAN====
Home Area Network:  uses cable, wired, or wireless connections to connect a homes' digital devices.  For example, fax machines, computers, DVD's etc. See [[w:Home network]].

Usually home networks work on P2P because one of them can't be spared to just be a server. Files are shared from each computer held in ''shared'' folders.

====BGAN====

Broadband Global Area Network supports mobile communications across an arbitrary number of wireless LANs and satellite coverage areas.
Example: Mobiles,internet etc..... [[w:Broadband Global Area Network]]

===P2P===
P2P (peer-to-peer) is a type of network where &quot;each workstation has equivalent capabilities and responsibilities&quot;[http://compnetworking.about.com].
*[http://ntrg.cs.tcd.ie/undergrad/4ba2.02-03/Intro.html Link to document on P2P networks.]

====MP3's====
MP3 is a format that allows audio (usually music) files to be compressed so they are small enough to be sent over the internet or stored as digital files. It is popular because the compression method takes out the sounds that less audible to humans thus retained as much sounds that can be heard by human ear in a relative small storage.

They are shared frequently on P2P networks on the internet. A portion of your hard drive becomes a server, so each P2P member is both a server and a client, both serving and receiving mp3 files. Napster is an example of a &quot;pure&quot; P2P network. Through this type of decentralized system, you are able to communicate from node to node, with out the use of a server.

[[Image:napster-logo.jpg|thumb|200px]]

===Internet===

The Internet is a worldwide, publicly accessible series of interconnected computer networks that transmit data by packet switching using the standard Internet Protocol. It is a &quot;network of networks&quot; that consists of millions of smaller networks, which together carry various information and services, such as electronic mail, online chat, file transfer, and the interlinked web pages and other resources of the World Wide Web (WWW). So the Internet is a collection of interconnected computer networks, linked by copper wires, fiber-optic cables, wireless connections, etc. In contrast, the Web is a collection of interconnected documents and other resources, linked by hyperlinks and URLs. The World Wide Web is one of the services accessible via the Internet, along with various others including e-mail, file sharing, online gaming and others.

[[w:Internet]]

====Intranet====
Intranet is a private computer network used by companies for employees.  It is only accessible within a limited area, thus increasing the security of the network.  It is not public and can't be accessed via WWW.

It can be described &quot;a private version of the Internet,&quot;
[[w:Intranet]]

====Extranet====
Extranets are basically intranets between a company and its as well as suppliers and customers who must login.

Extranets may be used for simple transactions such as purchasing and have become very popular.

====VPN====
Virtual Private Network shares wires with another network, but has encrypted packets of data that only you can see. It is private through technology, thus ''virtual''.
[[w:Virtual_private_network]]

==hardware==
===Topologies===
====Bus====
All nodes are connected to a single wire or cable (the bus) which has two endpoints.  Each communications device on the network transmits electronic messages to other devices.  If some of those messages collide, the sending device will wait and then try to transmit again.
The advantage of bus network is that it can be organized as a peer-to-peer network or client/server network.

It is also relatively inexpensive to install.

The disadvantage is that if the bus network fails, the whole system network fails.

The wider the bus the better!

====Ring====
The ring network is a network in which all communications devices and microprocessors are connected in a continuous loop. Electronic messages pass around the ring until they reach the correct destination; there is no central server.

The advantage of a ring network is that messages only flow in one direction.

The disadvantage is that if a single connection is broken, the whole network stops working.
The distributed star or tree topology can provide many of the advantages of the bus and the star topologies. It connects workstations to a central point, called a hub. This hub can support several workstations or hubs which, in turn, can support other workstations. Distributed star topologies can be easily adapted to the physical arrangement of the facility site.
(Integrated Publishing)

====Star====
A network that links all microcomputer and other communication devices through a central server.

The advantages of this hub is that it prevents impact between messages. The network forms a star shape having the central server in the middle with a single branch outward, the branch will not branch to another device.

Other Advantages Include:
* Good performance.
* Scalable, Easy to set up and to expand.
* Any non-centralised failure will have very little effect on the network, whereas on a ring network it would      all fail with one fault.
* Easy to detect faults
* Data Packets are sent quickly as they do not have to travel through any unnecessary nodes.
* It is used for centralised control.

===Components===
====Host====
A central computer (mainframe or midsize) that controls a network and the devices on it, called nodes.

The host computer has all the control over who has access to what hardware, software and all other resources on the network.
====Node====

A node is any device that is attached to a network.
Some examples of a node are: printer, terminal, microcomputer, and storage device.

====Hub====
A hub facilitates multiple input of the same device. It acts like a power bar to enable the multiple plugs to share 1 source of electricity. For example, there is a ethernet hub where all the ethernet cables are connected to to share a terminal.  A USB hub will allow multiple units of USB devices to plug into a single USB port.

A hub is called a half-duplex device because data can not be transfered back and forth simultaneously, it only goes one way at a time.

====Switch====

It connects a single computer to the network and allows it to use all the bandwidth  available. Unlike a hub, it sends messages to the computer that is the intended recipient.

Data can be transmitted back and forth at the same time, improving the performance of the network. This is called a full-duplex device.

Can be used in connection with a hub.

====Bridge====
Connects local area networks which are '''similar''' using a bridge interface to make a larger network. It also connects the same kind of networks This is crucial because similar networks can be joined together to make bigger ones!

====Gateway====

A gateway allows the communication between '''dissimilar''' networks. It can be between a WAN and a LAN or 2 LANs on different operating systems or layouts.  A gateway can come as either a hardware, software or possibly both.

====Router====
A router is a device that connects several devices together, and directs messages to communicate them.
This is important because high speed routers can handle major data traffic.

====Backbone====
As the name suggests, the backbone includes significant communication equipment such as gateways and routers that serve to connect computer networks within an organization.

The thing that distinguishes the '''backbone''' from other paths on the network is it's speed (bandwidth). The backbone is the fastest.
This is important because it is the central structure that connects all other elements of the internet.

===bandwidth===
Bandwidth refers to the amount of information that can be sent through a given communication channel in a given amount of time.
When connecting to the internet you have the option of connecting through:
*narrow bandwidth (100 kbps - kilo bits per second)
*medium bandwidth (1 Mbps)
*broad bandwidth (100 Mbps)

===wires===
====twisted pair====
A twisted pair is two copper wires that act as insulators that are &quot;twisted&quot; together. Its purpose is to reduce the interference that comes from electric fields. Twisted pair wire has been the most common channel (or medium) used for telephone systems.

====coaxial cable====
A coaxial cable consists of a center conductor surrounded by an insulator which is in turn surrounded by an outer conducting shield. Coaxial cables are used to carry high frequency (usually radio frequency)signals for long distances. It can be used for cable television.

====fiber optic====
Fiber optics is a glass or plastic fiber designed to guide light along its length.
Optical fiber can be used as a medium for telecommunication and networking. It currently supplies the widest commercial bandwith available.
[[w:Fiber_optic]]

===wireless===
====IR====
Infra red light can be used to transfer data wirelessly. This is how remote controls work. Line of sight must be preserved for it to work.
====RF====
Radio frequency is a combination of electrical energy and magnetic energy that carries communication signals. It is short for radio frequency and is measured in Mhz. Internationally, the RF spectrum is assigned by the International Telecommunications Union in Geneva, Switzerland.

====microwave====
Microwaves are electromagnetic waves with wavelengths shorter than one meter and longer than one millimeter, or frequencies between 300 megahertz and 300 gigahertz.
[http://www.wikipedia.org]
Microwaves can't curve or bend around the earth, so there has to be many microwave stations placed &quot;25-30 miles&quot; away from each other (line-distance), with nothing to block the connection. Each station takes signals from the next and makes the signal stronger and then sends it to the following station. (361)

===Short-range wireless===
====Bluetooth====
Bluetooth provides a way to connect and exchange information between devices such as mobile phones, laptops, PCs, printers, digital cameras, and video game consoles over a secure, globally unlicensed short-range radio frequency.
[[w:Bluetooth]]

====ZigBee====
It is a commerical network wireless protocol. It is low-cost, low-power, long battery life, wireless sensor networks. [http://www.freescale.com]

====Wi-Fi====
Open source wireless network protocols. We use it in college to connect wireless laptops.
WiFi has a lot of advantages. Wireless networks are easy to set up and inexpensive. They're also unobtrusive -- unless you're on the lookout for a place to use your laptop, you may not even notice when you're in a hotspot.
A wireless network uses radio waves, just like cell phones, televisions and radios do. In fact, communication across a wireless network is a lot like two-way radio communication. Here's what happens:
A computer's wireless adapter translates data into a radio signal and transmits it using an antenna.
A wireless router receives the signal and decodes it. It sends the information to the Internet using a physical, wired Ethernet connection.

====GEO====
[[File:Geostat.gif|thumb|200px|a picture of geo]]
Short for Geostationary Earth Orbit, a satellite system used in telecommunications. GEO orbit the earth at 22,300 miles above the earth's surface. They are tied to the earth's rotation and are therefore in a fixed position in space in relation to the earth's surface. The satellite goes around once in its orbit for every rotation of the earth. The advantage of a GEO system is that the transmission station on earth needs to point to only one place in space in order to transmit the signal to the GEO satellite. GEO systems are used for transmissions of high-speed data, television signals and other wideband applications.&lt;ref&gt;http://wi-fiplanet.webopedia.com/TERM/G/GEO.html&lt;/ref&gt;

[[w:Geostationary_orbit]]

====Piro====
[[File:Orbits around earth scale diagram.svg|thumb|200px|The green dotted line is the MEO orbit]]


Short for Middle Earth Orbit, a satellite system used in telecommunications. MEO satellites orbit the earth between 1,000 and 22,300 miles above the earth's surface. MEOs are mainly used in geographical positioning systems,known as GPS and are not stationary in relation to the rotation of the earth
Medium Earth orbit satellites (2000 km).
[[w:Medium_Earth_Orbit]]

====LEO====

Short for Low Earth Orbit, a satellite system used in telecommunications or data communication such as e-mail, paging and videoconferencing.&lt;ref&gt;http://wi-fiplanet.webopedia.com/TERM/L/LEO.html&lt;/ref&gt;

Low Earth Orbit satellites (400 km) used for communication. More satellites are needed (because fewer are visible around the horizon) but they are cheeper to get ''up'' and require less signal strength.
[[w:Communications_satellite#Low-Earth-orbiting_satellites]]

===NOS===
NOS stands for Network Operating System. NOS is software that manages the activity of a network through an operating system [[w:Network_operating_system]]

Some examples of popular NOS software are Novell NetWare, Linux, and Microsoft Windows NT/2000.

===packet===
[[Image:packet.png|thumb|200px|network packet]]
A packet is a formatted block of data that is carried by a packet mode computer network.
When data is converted into a packet, the network can give out longer messages, more reliably.
[[w:Packet_(information_technology)]]

===Protocol===

[[Image:Handshake.jpg|thumb|300px|one of the first stages of a protocol is to shake hands]]

A standard that controls the connection or communication between two end points of computing.
It can be controlled by hardware or software or a combination of the two. In other words, a protocol defines the behavior of the computer.

===ethernet===
[[Image:computer-connect-ethernet.jpg.jpg|thumb|200px|Ethernet]]
A local area network technology which can be used with almost any kind of computer. Describes how data can be sent in packets, within a range, between computers and other net worked devices. Ethernet is a family of frame-based computer networking technologies for local area networks.

===token ring===
[[File:Token ring.svg|thumb|200px|Token Ring Network]]
Token Ring is a local area network (LAN) where all computers are connected in a ring or star which a bit is used in order to prevent the collision of data between two computers that which would like to send messages at the same time.  Is it said that the Token Ring protocol is the &quot;second most widely-used protocol on local area networks after Ethernet&quot;.[http://searchnetworking.techtarget.com/sDefinition/0,,sid7_gci213154,00.html]

Source: http://searchnetworking.techtarget.com/sDefinition/0,,sid7_gci213154,00.html

==Examples==
===at home===
====ethernet====
====HPNA (phone)====
====HomePlug====
===GPS===
====24 MEO's====
====connect 4====
====10 feet accurate====

When downloading GPS signals via GPS receiver &amp; software you will get two answers - 1) broadcast ephemeris &amp; 2) precise ephemeris. This is due to an effect known as the Doppler effect. Similar to sound waves from a moving vehicle being heard after a car passes line of sight, satellites will pass before the radio signals reach the receiver on Earth. When you open your file with the downloaded signals you will, depending on the software, get two sets of data. One with a corrected set of coordinates that account for the velocity of the satellites, their instant coordinates &amp; the time it takes for their radio signal to reach Earth, known as Precise Ephemeris. The second set will contain the satellites' position at the time they broadcasted it from orbit, known as Broadcast Ephemeris.

===pagers===
====one-way====
====two-way====
====wireless email====
===cell phones===
====1G analog====
====2G digital====
====3G smart====
3RD GENERATION

====4G smart====
4th Generation

===References===
{{reflist}}

[[Category:Introduction to Computers]]

&lt;big&gt;Course Navigation&lt;/big&gt;
{| class=&quot;wikitable&quot;
! [[Introduction_to_Computers/Personal|'''&lt;&lt;  Previous - Personal''']]
!                                                        
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! [[Introduction_to_Computers/Security|'''Next - Security  &gt;&gt; ''']]
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Welcome to my page!! I am glad to be a wikiversitian,and very new here.
I heard that this site is the best school on the internet,I'm in college.
No [[Wikipedia:Administrators]] or anyone else from wikipedia can go to my page!!
That's why I have an account in Wikipedia,so Wikipedians can cooment.Only Wikiversitians
can comment me.
Wikipedia username:[[w:User:S495|S495]]
The page is to be continued until a later date</text>
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Welcome to my page!! I am glad to be a wikiversitian,and very new here.
I heard that this site is the best school on the internet,I'm in college.
No [[Wikipedia:Administrators]] or anyone else from wikipedia can go to my page!!
That's why I have an account in Wikipedia,so Wikipedians can cooment.Only Wikiversitians
can comment me.
Wikipedia username:[[w:User:S495|S495]]
The page is to be continued until a later date.</text>
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    <title>Going naked - Openism and freedom in academia</title>
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{| class=&quot;wikitable&quot; style=&quot;margin: 1em auto 1em auto&quot;
|-
| Take some time to look at this image... What is going on? What does the image say about ''{{font|color=red|sharing}}''?

''Some people {{font|color=red|shared}} some words and pictures in books. Children reading those books {{font|color=red|shared}} them with each other. Someone took a black and white photograph of these students {{font|color=red|sharing}} their books. That photograph was {{font|color=red|shared}} as a life-size print in a museum. Whilst a girl visiting the museum tried to look inside one of the books being {{font|color=red|shared}} (she didn't want to miss out!) another photographer took a photo - of the girl looking at the photograph of the children {{font|color=red|sharing}} the books ... This photograph was then {{font|color=red|shared}} electronically on flickr using an [[open license]] that allowed it to be imported to [[commons:|Wiki Commons]] and further {{font|color=red|shared}} on this page - you can use it too! ... and on it goes... ''[[Open academia]] is about {{font|color=red|sharing}} knowledge and learning resources in a mutually-enabling way - i.e., so that ideas can be re-mixed and re-iterated rather than coming to a dead-end. [[File:Gatunek_trujący.svg|20px]]
|}
{{center top}}[[File:GoingNakedOpenAcademiaSessionWordle.png|thumb|center|upright=3|{{center top}}[http://wordle.net Wordle] of this page.{{center bottom}}]]{{center bottom}}
{{RoundBoxLeft|theme=9|width=60}}
A 15-minute presentation about open academic philosophy and practice.

By [[User:Jtneill|James Neill]] - with responses by [http://mlearning.edublogs.org Leonard Low]&lt;!-- :&quot;[[Open Slather]]&quot; --&gt;, [[User:Leighblackall|Leigh Blackall]]&lt;!-- , [[User:Madepercy|Michael de Percy]], etc. (add yourself)) --&gt;.

'''Where''': Hothouse, 1C32, [[University of Canberra]]

'''When''': Friday 5th March, 2010 13:30
&lt;!--
'''Ustream''': http://www.ustream.tv/channel/hothouse-seminar-series --&gt;

''Now a solid(ish) draft - but still feel free to improve or comment''
----
'''Summary''': [http://leighblackall.blogspot.com/2010/03/going-naked-openism-and-freedom-in.html Going Naked - Openism and Freedom in Academia] (Leigh Blackall blog)

'''Also recommended''': [http://www.slideshare.net/opencontent/wiley-slidesv5 Open education] slides by David Wiley, 2010
----

[[File:Unicode 1xF12F.svg|center|frameless|upright]]
The [[w:copyleft|copyleft]] symbol mirrors the [[w:copyright symbol|copyright symbol]] to represent freedom and openness in licensing [[w:intellectual property|intellectual property]]. See also [[free culture]].
{{RoundBoxRight|theme=9}}
{{TOCright}}
  {{RoundBoxBottom}}
==Screencast==
[[File:GoingNakedOpenismAndFreedomInAcademia.ogv|center|thumb|upright=2|A 16 minute screencast explaining the concept of [[open academia]], its principles, and steps towards practice. You can also watch on youtube: [http://www.youtube.com/watch?v=IFai7iGEw_A Pt 1] (8 mins) and [http://www.youtube.com/watch?v=IFai7iGEw_A Pt 2] (8 mins).]]

==Abstract==
&lt;!--- Images ---&gt;
[[File:Barn raising - Leckie's barn completed in frame.jpg|thumb|right|Knowledge-building is a team effort - you help build mine, I help build yours; what goes around, comes around.]]
[[File:Escape the ivory tower.jpg|right|thumb|To what extent do you  escape the [[w:ivory tower|ivory tower]] and participate in [[w:the commons|the commons]]?]]
[[File:William-Adolphe Bouguereau (1825-1905) - Thirst (1886).jpg|thumb|right|People are
thirsty for open academia.]]
[[File:Book 06490 20040730160049 L.jpg|right|thumb|[[w:Textbook#The &quot;broken market&quot;|Traditional textbooks are a &quot;broken market&quot;]], hence driving [[w:open textbook|open textbook]] alternatives.]]
&lt;!--- End images ---&gt;
[[w:Academia|Academia]] should be conducted in such a way as to benefit society. This means (among other things) that the processes and products of publically-funded academics' activities should, by default, be public (i.e., accessible and freely usable). It also means that academics should seek to use and promote tools (such as [[software]]) and materials (such as [[w:textbook|textbooks]]) which enable &lt;del&gt;students&lt;/del&gt; [[emerging academic]]s to utilise and foster public knowledge. The evolution towards open academia is a cultural challenge because closedness is the norm.

==Academia is about sharing==
&lt;blockquote&gt;{{font|color=purple|face=Tahoma|&lt;big&gt;If I give you a penny, you will be one penny richer and I’ll be one penny poorer. But if I give you an idea, you will have a new idea, but I shall still have it, too.&lt;/big&gt;&lt;br&gt;
 - [[w:Albert Einstein|Albert Einstein]]}}&lt;/blockquote&gt;

&lt;blockquote&gt;{{font|color=purple|face=Tahoma|&lt;big&gt;He who receives ideas from me, receives instruction himself without lessening mine; as he who lights his taper at mine receives light without darkening me.&lt;/big&gt;&lt;br&gt;
- [[w:Thomas Jefferson|Thomas Jefferson]]}}&lt;/blockquote&gt;

Academia is about sharing - otherwise it is not academia. To not share is to retreat to an [[w:ivory tower|ivory tower]]. This is the &quot;low game&quot; of academia in which knowledge-development and knowledge-storing is approached as a competition (e.g., between staff, students, departments, institutions, sectors, countries etc.). The &quot;higher game&quot; in academia is to selflessly contribute to collective knowledge by freely disseminating one's knowledge and activities (for a deeper discussion of the &quot;academic game&quot;, see De Ropp (1968)&lt;ref&gt;De Ropp, R. S. &amp; Lourie, I. (2003). ''[http://books.google.com/books?id=JScDAAAACAAJ The master game: Pathways to higher consciousness]'' (3rd ed.). Gateways Books &amp; Tapes.&lt;/ref&gt;).

This claim may be summarised as &quot;academics are public servants and our work is public property&quot;.

&lt;!-- Images ---&gt;
[[File:Bertall ill Les Habits Neufs du Grand Duc duc.png|right|thumb|[[w:The Emperor's New Clothes|The emperor in his new clothes]]. Academics could be similarly attired so we can see what they really offer society. Could this be a consequence of increasing demand for public scrutiny and transparency of university performance? e.g., [http://www.theaustralian.com.au/higher-education/after-my-school-comes-my-university/story-e6frgcjx-1225836275192] ]]
&lt;!-- End images --&gt;

==The default is (becoming) open==

The cultural standard for public institutions is increasingly moving towards a &quot;default is open&quot; philosophy and practice. Several business sectors are also participating in this evolution towards openness.

Openness-closedness is not a dichotomy - it is a continuum - arguably, an evolutionary continuum.

To not engage with the palpably shifting tide towards openness is risky for academic institutions (e.g., funding metrics are likely to become more accurate at measuring the social value&lt;ref&gt;O'Driscoll, T. (2010). [http://opensource.com/business/10/2/social-production-new-source-economic-value-creation Social production as a new source of economic value creation] opensource.com.&lt;/ref&gt; generated by academic institutions). Even more importantly, democratic academic institutions should be leading the charge towards openness since innovatively fostering public knowledge is their raison d'etre.

The biggest barriers to openism in academia are not legal or technical, but rather cultural, organisational, and psychological (because closedness is the norm). Most academics, for example, use closed textbooks and websites for teaching, publish in closed rather than [[open journal]]s, user proprietary software, and do not make their work openly available.

==5 pillars of open academia==
&lt;blockquote&gt;{{font|color=purple|face=Tahoma|&lt;big&gt;If you focus your mind on the freedom and community that you can build by staying firm, you will find the strength to do it.&quot;&lt;/big&gt;&lt;br&gt;
 - [[w:Richard Stallman|Richard Stallman]]}}&lt;/blockquote&gt;

In practice, going naked (free and open) can be a surprisingly complex and challenging endeavour - mainly because of the widespread cultural habits of closedness which need to be unlearnt. It can also be a liberating and empowering journey. By &quot;going naked&quot; and not carrying around the heavy weight of &quot;closedness&quot; much of one's energy and focus is freed. 

Culturally, what's needed is to bulldoze the ivory tower and replace it with an open parthenon consisting of at least five pillars. These &quot;five pillars of open academia&quot; offer a practical, guiding framework for moving towards openism:

[[File:5 pillars of open academia.svg|center|500px]]

Expressed as a &quot;creed&quot;, 
&lt;blockquote&gt;{{font|color=green|&quot;as an open academic I commit to providing '''open access''' to all my academic outputs (teaching, research and service) using '''open formats''', '''open licensing''' and '''free software'''. I also commit to '''open management''' of my academic activities.&quot;}}&lt;/blockquote&gt;

==Open university==
[[File:BallonKathedrale01 edit.jpg|right|thumb|180px|A mobile, free-knowledge wifi? Symbol of an [[open university]]?]]
A closely related conceptualisation is provided by the Free Software Foundation (FSF; initiated by Students for Free Culture (SFC)). They define an [[open university]] as one in which:
# The research produced is [[open access]];
# The course materials are [[open educational resources]];
# The university embraces [[free software]] and [[open standard]]s;
# The university’s patents are readily licensed for free software, [[w:essential medicine|essential medicine]], and the [[w:public good|public good]];
# The university’s network reflects the open nature of the [[w:Internet|Internet]]. 

The FSF goes further, to list specific criteria for each of these open university principles and university-rankings based on their degree of openness: [http://wiki.freeculture.org/Open_University_Report_Cards Open University Report Cards].

Increasing transparency in higher education is currently being strongly by the Labour government in Australia. On the 3rd March, 2010 the government announced the creation of a &quot;My University&quot; federal website by 2012 (based on the &quot;My School&quot; site) which will provide transparent, comparative data about indicators of university quality (such as student satisfaction ratings). In response the chief executive of Universities Australia, Glenn Withers, welcomed the idea:

&quot;Universities are fully committed to transparency. They are remarkably open already via their websites, public guides, reports to parliament, auditors general, ombudsmen and more. They fully welcome anything that can enhance this transparency through new, well-designed initiatives for universities and importantly, for all other tertiary providers.&quot; (Harrison, 2010&lt;ref&gt;Harrison, D. (2010). [http://www.smh.com.au/national/education/my-school-for-universities-on-the-way-20100302-pges.html My School for universities on the way. ''Sydney Morning Herald''], ''March 3''&lt;/ref&gt;)

==Making the change==
&lt;blockquote&gt;{{font|color=purple|face=Tahoma|&lt;big&gt;If you want to accomplish something in the world, idealism is not enough - you need to choose a method that works to achieve the goal.&lt;/big&gt;&lt;br&gt;
 - [[w:Richard Stallman|Richard Stallman]]}}&lt;/blockquote&gt;

Two methods of moving towards greater openism in academia might be:
# Cold turkey (e.g., new institutions and projects and institutions undergoing major change may simply adopt an open charter from the outset)
# Iterative/progressive (e.g., for existing institutions, openist [[w:Key performance indicator|KPI]]s can be established and made part of their strategic plans and roadmaps can be developed (e.g., [http://www.olcos.org/cms/upload/docs/olcos_roadmap.pdf])
# Do a cost/benefit analysis measuring the costs of closed, including software, login caused inefficiencies and administration, copyright royalty costs, against the benefits such as free media, free software, skills, productivity, social capital, marketing benefits..

==Activity==
What are your thoughts about '''[[open academia]]'''? Helpful ways to structure responses could be in terms of barriers and benefits, and/or a [[w:SWOT analysis|SWOT analysis]] - just click edit, add your thoughts, and save {{smile}}:

===Barriers and benefits===
What barriers are there to open academia? What benefits might there be?
====Barriers====
* Performance anxiety
* Control and ownership
* Ignorance
* Time/prioritisation, full disclosure and informed choice

====Benefits====
* Moral: Helps to fulfill the charter of public universities to contribute to the common good of society

===SWOT analysis===

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===Strengths===
* Addresses copyright ambiguity for educational use
* Enables reusability 
* Liability is individualised
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===Weaknesses===
* Individually simple, organisationally difficult
* Difficult to implement with so much of media commercially orientated, so formats and devices often don't record or play open formats 
* Skills shortage: Computing and IT sector has been trained to service commercially developed software, hardware, media (CISCO, Microsoft).
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===Opportunities===
* A principled stance against [http://leighblackall.blogspot.com/2009/12/developing-copyright-policy-at-uc.html the questionable academic publishing norm]
* [http://delicious.com/leighblackall/-feedback-otagopolytechnic Increase the profile] of UC at this time, as being a leader in the field, recognising the problems and hypocrisy, taking a principled stance
* [http://wikieducator.org/Otago_Polytechnic/Measuring_our_open_education Good economic returns] through marketing and profile gains, legal and administrative savings, infrastructural savings, efficiency and sustainability, and skills capacity building.
 
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===Threats===
* The [[w:Status quo|status quo]] favours closedness 
* Commercial appropriation (and &quot;enclosure&quot;) of freely available research
* Cultural imperialism - forcing copyright restrictions on people with one hand, then swamping then with open educational media
* Loosing or weakening whatever stake we in academia and education might have had with Fair Use/Dealings
* &lt;!-- responses here --&gt;
* 
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==Responses==
Feel free to share your thoughts, ideas, resources, and experiences by clicking '''edit''' - or use the [[Talk:Going naked - Openism and freedom in academia|talk page]].

----

Completely agree with the ethos of course.  I'm interested in the assertion that other parts of society are becoming more open - I'm not convinced.  Knowledge is horded by elites as a source of power, and as such is hidden by a variety of mechanisms, eg legal, technocratic language, by omission.  It seems to me that universities pretty much invented the 'knowledge economy' - that is the commodity they trade in.  They package it up into 'degrees', they market their 'unique' courses to prospective students, they create competition for knowledge by limiting access, and then by grading, rather than simply critiqing, students' work.  The only point at which they &quot;gift&quot; knowledge is the publication of academic work - and they have managed to commodify that by linking publication to prestige (read market positioning of individuals and institutions)

University 'openism' is truly counter-cultural in that it means that universities get out of the business of simply re-creating elites.  This would indeed be a revolution.  So two lines of thought occur
# What then is the role of universities?... are they simply a trusted source of knowledge?  Does this not still link to elitism?  And who is funding this &quot;public service&quot; (180 degrees away from current corporatisation of unis)
#  Have we then delivered an egalitarian utopia... or do elites simply colonise another structure - and what would that be?

cheers, Jane

: Thanks for finding the &quot;edit&quot; button and daring to share, Jane. Your comments have me pondering further, thank-you. Is the rest of society becoming more open? In making this suggestion, I'm thinking e.g., of the steps being made towards open government (making public data open by default), of p2p networks sharing (which no-one really can stop it seems), of the uptake of open source software (e.g., Firefox) etc. Maybe it is rose-coloured glasses from me - there is plenty of growth in closed/patented/copyrighted efforts too. But maybe both are true - '''maybe we are becoming simultaneously more open and more closed - that could be possible?'''

: It seems to me, yes, as you say, universities have created (or at least fostered) a 'knowledge economy' to justify/maintain their existence - and lost focus on their social role/purpose - at least the university 'charters' don't seem to match the behaviour. And if universities can't/don't/won't do openism (which is their potential social value) then knowledge-brokering is in big trouble - how else can we expect openism in other facets of society to flourish if the institutions supposedly set up for knowledge-sharing are lapsing into small-minded protectionism? So the &quot;egalitarian utopia&quot; seems a long way off (but then, so do most utopias) - the multinationals have swallowed the textbook and journal industry and are busy clawing their fingernails into classrooms, learning management systems, etc. with academics seduced like doctors by pharmaceutical companies... but '''we can &quot;subvert the dominant paradigm&quot;''' - at least that was the graffiti I remember from my undergrad ways. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 13:38, 4 March 2010 (UTC)

:: I think we are becoming simultaneously more open and more closed. I think a 'knowledge war' of sorts exists. People feel knowledge is a right and simultaneously feel they have a right to profit from knowledge. I think polarization is pretty common in most things. When people cannot find common ground polarizations turns into extremes. I think detecting extreme polarizations is easy because terms like left, right, liberal, conservative. etc. tend to get thrown around, and people start identifying themselves and others by such labels. I won't be surprised when people start saying &quot;I am a knowledge liberal&quot; or &quot;you are a right extremist on the knowledge issues&quot;. Just some thoughts, maybe this is not really on topic. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 02:05, 5 March 2010 (UTC)

This is an interesting discussion.  Based on the above thoughts, here's my new hypothesis.  That individuals are becoming more open, due to the symbiotic relationship between technological advances and cultural changes to do with personal freedom and privacy issues. The individual is free to be altruistic and share knowledge. But this altruism does not translate well into the institutional/corporate sphere (at best partially, after a risk management process).  The self-interest of the institution/corporation/professional groups (elites) is to maintain competitive advantage through ownership and control of knowledge resources (including education).  

The university sector becomes the interesting case, as the institutional ideals clash with institutional self-interest.  So I think your ideas around re-examining and reframing the institutional self-interest are practical... potentially 'subverting the dominant paradigm'??!!  I guess the question is whether the university sector can be a master of its own destiny, or whether it is a tool of other dominant forces.  cheers Jane

: I tend to agree with some of the basic concepts in your new hypothesis. Knowledge is a natural resource like water. I think individuals can have both a personal self-interest and a professional self-interest in knowledge. The personal self-interest of individuals is to want free, open, and easy access to knowledge and the freedom to use and share knowledge. The professional self-interest of individuals is usually driven by a basic social need to have money to pay for other resources, like water. Personal self-interest looks for ways to make knowledge freely accessible and easily obtainable in order to have the freedom to use and share knowledge. Professional self-interest looks for ways to drive and/or control supply and demand, just like with water, in order to maintain itself and pay workers. I think transition between the personal and the profession (or visa versa) is naturally assumed to be incompatible because individuals often do not understand the driving forces or the motivations of the personal and professional parts of individuals. This is probably what leads to the idea that individuals or groups have bad intentions, and attempts to protect the personal and professional self-interests.

: If you look and understand how the copyleft model manages to work, I think you may be able to see how both can work together. Universities can sell field experience and teachers as knowledge support, along the same lines as 'tech support' — in fact 'tech support' is a form of knowledge support because people seek technical knowledge on how to do or fix things. In this model knowledge is a resource that can be used and shared freely, and Universities offer experience and knowledge support as their services that people can buy. The change you talk about involves a move away from knowledge as an exploitable resource to a knowledge services support industry. I think a natural fear of change, the uncertainty and risk that comes from not knowing what would happen and how change would work, is usually what keeps individuals from pursuing change. I think professional change usually comes from necessity and from entrepreneurs. I think necessity comes into play when the majority of consumers refuse to buy the product being sold which in this case would be knowledge. When obtaining knowledge is no longer a major driving force in what University individuals decide to attend Universities will change by necessity. I think entrepreneurs will first provide alternative educational models that appeal to the majority of consumers before Universities catch up. Although there may be some intelligent individuals that see a trend and make changes to the University model before the majority of consumers want something else. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 10:01, 5 March 2010 (UTC)

If history is any indicator, I think things will get worse before getting better. I can see a scenario where Universities become scared of their students undermining there business model and manage to control and limit access to knowledge by finding a way to make it illegal for students to share what they learned with others. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 10:22, 5 March 2010 (UTC)

Some more random thoughts. In [[#Abstract|Abstract]] it is said that &quot;Academia should be conducted in such a way as to benefit society.&quot; In [[#Academia is about sharing|Academia is about sharing]] it is said that &quot;Academia is about sharing - otherwise it is not academia.&quot; I think an assumption is being made that these two statements are true, and everybody knows this to be true is taken for granted. I think anyone trying to learn from this page could benefit from understanding why academia should benefit society and why academia is about sharing and why if there is no sharing it is not academia. --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 02:20, 8 March 2010 (UTC)
: Agreed - this is in need of expansion/development/argument. Curious, - what do you think of [[w:Academia]] - e.g., consistent or not so consistent with these claims - or itself in need of such development? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:32, 8 March 2010 (UTC)

:: I do not know enough about how things work at Wikipedia to comment on development there. I mostly just read Wikipedia, and I usually don't read an entire article because I think Wikipedia's articles usually suffer from information overload. I sometimes end up turning to Simple Wikipedia in order to get quicker answers to what I would like to know. What specifically about [[w:Academia]] should I be looking at to tell if its consistent or not so consistent with these statements? --[[User:Darklama|&lt;span style=&quot;background:DarkSlateBlue;color:white;padding:2px;&quot;&gt;&amp;nbsp;dark&lt;/span&gt;]][[User talk:Darklama|&lt;span style=&quot;background:darkslategray;color:white;padding:2px;&quot;&gt;lama&amp;nbsp;&lt;/span&gt;]] 02:48, 8 March 2010 (UTC)
::: No worries - I'll take a closer look - the quick gist I got was that it seemed at least loosely consistent with academia as characterised by sharing, but I need to read/think more in this area to articulate such a position. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:24, 9 March 2010 (UTC)

===Some responses from Leonard Low and Leigh Blackall===
'''Currently as noted by [[User:Jtneill|James Neill]] and [[User:Leighblackall|Leigh Blackall]]'''
* Leonard: There is an increasingly litigious intellectual property culture - therefore are we to play into the hands of commercial interests by releasing our material (for their use and gain/benefit)?
* Leonard: Is it a university's duty to give back to society at large or to our students in particular? Leonard raises the issue that if a university makes all of its learning freely and accessibly available what advantage is there for paying students?
* Leigh: A Share Alike restriction prevents exploitation of openism?
* Leonard: If the knowledge is freely available, does this mean that a university is just a rubber-stamping, degree-granting institution?
* Leonard: What's the advantage of going to university if the info is readily available?
* Leigh: The value proposition of a university education is assessment and certification, not content.
* James: A basic business model for an open university is that it:
*# provides training, support, guidance, teaching, assessment, feedback, and grants degrees etc. to paying students (funded publically or privately)
*# conducts research (funded publically or privately)
*# freely and openly shares its knowledge (teaching, research, and service) and activities (e.g., open management) - otherwise it is a private college, not a university
* Leonard: Ivory tower is about engagement with society (means academics who do not engage with society) - one could still be in an ivory tower and open if the academic activity is not engaged with society's needs (i.e., relevance of content matters too - not just openness)
* Leigh: Argues that the irrelevance of academic pursuits becoming &quot;ivory towers&quot; should be less of an issue through the Internet and can be addressed though popularisation - such as engaging in Wikimedia Foundation work, publishing video synopsis on Youtube, and networking online. See Using the [http://www.slideshare.net/leighblackall/popular-internet-in-teaching-and-research-2631841 Popular Internet in Teaching and Research] 
* Leonard: On the balance of things, Leonard personally supports openness
* Leigh: The risk of open education becoming a force for neo colonialism. While on the one hand our trade agreements impose copyright regulation on countries lacking resources to produce their own content, in the other we make our a selection of our own content free of use. At the moment, free content tends to be US centric for example. See [http://leighblackall.blogspot.com/2009/08/looking-into-sky-open-ed-oh-nine.html New Colonialism in OER]
* Leigh: How much does UC pay [http://www.copyright.com.au/Copyright_Users/Education/Education.aspx Copyright Agency Limited (CAL)] for generalised royalty fees for the copying of restricted content? How much could be saved if a Faculty were to declare only using free and open content?
* Leonard: [http://www.copyright.com.au/Copyright_Users/Education/Education.aspx CAL] fees are in the millions per year
* Leigh: Regarding UC's copyright IP policy, and its recent review status. Leigh attempted to engage with UC leadership on the review but was advised that procedure requires only external agencies review UC policy. Leigh wants to know why this is and who the agency was, and what their advice was.

==Proposed actions==
* The [[University of Canberra]] Teaching and Learning Centre to find out the annual fees paid to [http://www.copyright.com.au/Copyright_Users/Education/Education.aspx CAL], and if possible - a break down of that fee.
* The [[University of Canberra]] Teaching and Learning Centre to find out why external agencies review UC policy, what agency reviewed UC's IP policy, and what their review was.
* Leigh and James to run Workshops on how to source free content and reuse it in a free way
* Leigh and James to propose the [[University of Canberra]] Faculty of Health lead the university and adopt a default open education and research position.
{{RoundBoxBottom}}

==References==
{{reflist}}

==See also==
* [[Using social media for teaching and research]] - the broader series of discussions
* [[Open academia]]
* [[Open access]]
* [[Open formats]]
* [[Open licensing]]
* [[Free software]]
* [[Open management]]
* [[User:Jtneill/Presentations/Open academia: A philosophy of open practice|Open academia: A philosophy of open practice]]

==External links==
* [http://www.theaustralian.com.au/news/nation/strip-off-for-photographer-spencer-tunick-at-the-sydney-opera-house/story-e6frg6nf-1225835469917 5200 strip off for photographer Spencer Tunick at the Sydney Opera House]: &quot;&quot;It doesn't feel sexual, it just feels tribal, a gathering of humanity.&quot;
* [http://opensource.com/law/09/12/ip-another-bubble-about-burst-view-another-civilization Is IP another bubble about to burst? A view from another civilization.], Venkatesh Hariharan, 16 Dec 2009
* [http://www.youtube.com/watch?v=52Gri8y9iYA Open educational resources at Otago Polytechnic] (Sunshine Connelly, 2009): Interviews with educators (8 min.) 
* [http://www.youtube.com/watch?v=J-63D6h0e38 Open source way] (30 sec video promo): &quot;What if the work of one became the mission of many? Maybe you could try this...&quot;
* [http://www.theaustralian.com.au/higher-education/who-will-own-what-we-read/story-e6frgcjx-1225833593341?from=public_rss Who will own what we read?], Colin Steele, Feb 24, 2010, ''The Australian''

[[Category:Open academia]]
[[Category:University of Canberra/Workshops]]
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      <text bytes="25244" sha1="6un1o5hhwjapxukr59lvmtbus3vw0xn" xml:space="preserve">== Introduction ==


[[Wikipedia:University|Universities]] are still the major place in society where, worldwide,
millions of people '''create, process, use, adapt and distribute knowledge''', in various forms and types. 
In their [[Wikipedia:Mission_Statement|mission statements]] universities often underline their [[Wikipedia:Public|public]] role in terms of the responsibility for active   [[Wikipedia:Knowledge_Sharing|knowledge sharing]] between [[Wikipedia:Academics|academics]] and the dissemination of knowledge to the society in general.
The '''‘public’''' role of the university can also be connected to its being a kind of “refuge” for research and education (K.U.Leuven speaks about vrijplaats) which obviously refers to its particular (academic) freedom.
In this contribution, however, public refers to the open character of generated knowledge and technology, though one could argue that
current policies at the  [[Wikipedia:Kuleuven|K.U.Leuven]] also limit  [[Wikipedia:Academic_Freedom|academic freedom]] and the notion of public that is implied.

Our aim is to focus on the tension between the overall ambition of an increased number of universities for '''active knowledge sharing'''
and the fact that their concrete policy practices rather '''discourage knowledge sharing''', even encourage knowledge [[Wikipedia:Privatisation|privatization]]. 
This tension will be illustrated by focusing on some concrete policy practices of the K.U.Leuven. 
More in particular we will clarify that the [[#Performance Indicators|performance indicators ]] that are formulated by the K.U.Leuven encourages researchers to generate knowledge and technology only under [[Wikipedia:Proprietary_Software|proprietary licenses]], that is, licensed under exclusive legal right of the copyright holder. 
In opposition to this, we will elaborate in the second part of our contribution on several methods to publish knowledge and technology under [[#Open licenses and open standards|open licenses]] and [[#Open licenses and open standards|open standards]]. 
These methods are not valorized and, as a result, discouraged by current policy practices at the K.U.Leuven also limit academic freedom and
the notion of public that is implied.

== Performance Indicators ==

In order for an organization to valorize its performance, the K.U.Leuven underlines that performance indicators are required. These indicators are formulated for research, academic service, education, internationalization, … etc. We focus here on the first two elements: research and academic service (wetenschappelijke dienstverlening).
The complete mission statement (only in Dutch)  
&lt;ref&gt;K.U.Leuven mission statement [ http://www.kuleuven.be/overons/pdf/academische_strategie_KULeuven.pdf], accessed December 6, 2010.&lt;/ref&gt; summarizes Key performance indicators (KPI) that are used to monitor the input, process and result. 

=== K.U.Leuven performance indicators for research ===

{| border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;center&quot;
|-
 ! style=&quot;background: #efefef;&quot; | KPI  for research 
|-
| Number of publications (mostly closed)
|-
| Impact factor for each publication 
|-
| Number of citations
|}

It is important to underline that the [[wikipedia:Kuleuven|K.U.Leuven]], by taking the number of published papers together with their [[wikipedia:Impact_factor|impact factor]] and number of [[wikipedia:Citations|citations]] as indicators for the valorization of research performance, imposes a policy that discourages active knowledge sharing. Researchers are stimulated to distribute their work in journals that come with high [[wikipedia:Impact_factor|impact factors]], yet [[wikipedia:Proprietary_licenses|proprietary license]]. Open knowledge sharing through open software, open content  or the development of [[# Open licenses and open standards|open standards]] simply do not count (no valorization) as publications. Most Open access journals, being only recently set up, are hardly valorized as they come with a very low impact (taking the citations of the last two to five years into account). As a consequence, these journals do not attract the best articles and there the viscous circle starts. In other words, the use of the K.U.Leuven performance indicators discourages people to openly exchange knowledge and technology and cultivates its privatization. This might sound strange in the context of a publication policy.

=== K.U.Leuven performance indicators for academic service (wetenschappelijke dienstverlening) ===

{|  border=&quot;1&quot; cellpadding=&quot;5&quot; cellspacing=&quot;0&quot; align=&quot;center&quot;
|-
!  style=&quot;background: #efefef;&quot; | KPI  for academic service   
|-
| Number of generated patents 
|-
| Number of generated spin-offs 
|-
| Industrial revenue (in euros) 
|}


Here again, we want to underline that, taking the number of generated [[wikipedia:Patent|patents]] or [[wikipedia:Research_spin-off|spin-offs]] as performance indicators, the K.U.Leuven discourages active knowledge sharing and supports knowledge privatization. Here as well, in view of the expectation to maximize patenting opportunities, researchers are stimulated to generate knowledge and technology under proprietary licenses. In this context, K.U.Leuven has set up a [[wikipedia:Technology_transfer|technology transfer]] office (TTO), the K.U.Leuven Research &amp; Development [http://lrd.kuleuven.be/ (LRD) ] that supports researchers in their interaction with industry and society (covering contract and collaborative research, patenting and licensing and spin-off creation) and in the valorization of their research results. The third indicator, industrial revenue, discourages researchers to use (mainly free) [[wikipedia:Open-source_software|open source software]] as they are rewarded for spending a lot of money (for example on software licenses).

==Open licenses and open standards==

In this short exploration of active knowledge sharing and open knowledge construction, we make a distinction between open software, open content, open access and open standards.

=== Open software licenses===

In contrast to proprietary software licenses that imply that certain rights regarding the software - inspection of code, modification and distribution - are reserved by the software publisher, [[wikipedia:Open_Software_License|open software licenses]] make software free for inspection of its code, modification, and distribution. These initiatives have the intention to maximise the distribution of knowledge that is collected in the software code. We will focus here on three distinct types of software licenses: 

# [[Wikipedia:GPL|GNU General Public License]] (GPL)
# [[Wikipedia:Berkeley Software Distribution|Berkeley Software Distribution]] (BSD) license
# [[Wikipedia:LGPL|GNU Lesser General Public License]] (LGPL)

Roughly speaking, the GPL forces users to release all their changes to GPLed software, including the software they added &quot;externally&quot;. The BSD does not impose any release of changes, but does not allow you to call the software your own. The LGPL is somewhere in between: it expects release of improvements, but allows [[wikipedia:Linker_(computing)| linking ]] with the software without the obligation to release it.

==== GNU General Public License ([[wikipedia:GNU_General_Public_License|GPL]]): Free speech as a moral duty ====

{{Infobox person
 |name          = Richard Matthew Stallman
 |image         =Richard_Stallman_at_Pittsburgh_University.jpg 
 |caption       = Richard Stallman at the [[wikipedia:University_of_Pittsburgh|University of Pittsburgh]] 2010 
 |birth_place   = [[wikipedia:New_York_City|New York City]], [[wikipedia:New_York|New York]], [[wikipedia:United_States|United States]]
 |known_for     = [[wikipedia:Free_software_movement|Free software movement]], [[wikipedia:GNU|GNU]], [[wikipedia:Emacs|Emacs]]
 |occupation    = President of the [[wikipedia:Free_Software_Foundation|Free Software Foundation]]
 |website       = [http://www.stallman.org www.stallman.org]
}}

'''[[wikipedia:Richard_Matthew_Stallman|Richard Matthew Stallman]]''' was the first important American ''''software freedom activist'''. 
In September 1983, he launched the[[wikipedia:GNU Project| GNU Project ]]&lt;ref&gt;{{cite web
|url=http://www.gnu.org/gnu/initial-announcement.html
|title=Initial GNU announcement
|date=1983-09-27
|accessdate=20 November 2008
|last= Stallman
|first= Richard
}}&lt;/ref&gt; to create a free [[wikipedia:Unix-like|Unix-like]] operating system, and has been the project's lead architect and organizer. With the launch of the GNU Project, he initiated the [[wikipedia:free software movement|free software movement]]; in October 1985 he founded the [[wikipedia:Free Software Foundation|Free Software Foundation]].

Stallman pioneered the concept of [[wikipedia:Copyleft|copyleft]] and he is the main author of several copyleft licenses including the [[wikipedia:GNU General Public License|GNU General Public License]], the most widely used [[wikipedia:free software license|free software license]].&lt;ref&gt;{{cite web
|url=http://www.dwheeler.com/essays/gpl-compatible.html
|title=Make Your Open Source Software GPL-Compatible. Or Else.
|accessdate=20 November 2008
|last= Wheeler
|first= David A.
|date= 2008-10-03
|work=(See the list in section 2)
}} &lt;/ref&gt;

[[File:Unicode 1xF12F.svg|thumb|left|alt=Letter c surrounded by a single line forming a circle.|The &quot;reversed 'c' in a full circle&quot; is the copyleft symbol. It is the [[wikipedia:copyright symbol|copyright symbol]] mirrored. Unlike the copyright symbol, it has no legal meaning.]]
'''Copyleft''' is a play on the word ''[[wikipedia:Copyright|copyright]]'' to describe the practice of using copyright law to offer the right to distribute copies and modified versions of a work and requiring that the same rights be preserved in modified versions of the work.  In other words, copyleft is a general method for making a program (or other work) free, and requiring all modified and extended versions of the program to be free as well.&lt;ref&gt;{{Cite web|url=http://www.gnu.org/copyleft |title=What is Copyleft? |accessdate=2010-08-29}} &lt;/ref&gt;

Copyleft licenses are sometimes referred to as &quot;[[wikipedia:Viral_license|viral license]]s&quot; because any works derived from a copyleft work must themselves be copyleft when distributed (and thus they exhibit a [[wikipedia:Viral_phenomenon|viral phenomenon]]). The term 'General Public Virus', or 'GNU Public Virus' (GPV), has a long history on the Internet, dating back to shortly after the GPL was first conceived.&lt;ref&gt;{{Cite web|url=http://psg.com/lists/namedroppers/namedroppers.2006/msg00246.html |title=Re: Section 5.2 (IPR encumberance) in TAK rollover requirement draft |first=Paul |last=Vixie |authorlink=Paul Vixie |publisher=[[Internet Engineering Task Force|IETF]] Namedroppers mailing list |date=2006-03-06 |accessdate=2007-04-29 |archiveurl = http://web.archive.org/web/20070927175628/http://psg.com/lists/namedroppers/namedroppers.2006/msg00246.html |archivedate = 2007-09-27}}&lt;/ref&gt;&lt;ref&gt;{{Cite web|url=http://catb.org/esr/jargon/oldversions/jarg221.txt |title=General Public Virus |work=[[Jargon File]] 2.2.1 |date=1990-12-15 |accessdate=2007-04-29}}&lt;/ref&gt;&lt;ref&gt;{{Cite journal|url=http://devlinux.org/lw-gnu-published.html |title=Reverse-engineering the GNU Public Virus&lt;/ref&gt;

&lt;p&gt;
Stallman argues that software users should have the '''freedom to share with their neighbour''' and to be able to study and make changes to the software that they use. He maintains that attempts by proprietary software vendors to prohibit these acts are '''antisocial and unethical'''. The phrase &quot;software wants to be free&quot; is often incorrectly attributed to him, and Stallman argues that this is a misstatement of his philosophy.&lt;ref&gt;[http://www.groklaw.net/article.php?story=20050513135545766 The Daemon, the GNU and the Penguin] by Peter H. Salus. Retrieved 18 February 2005.&lt;/ref&gt; He argues that freedom is vital for the sake of users and society as a '''moral value''', and not merely for pragmatic reasons such as possibly developing technically superior software.
&lt;/p&gt;
&lt;br&gt;
&lt;br&gt;

====  Berkeley Software Distribution ([[wikipedia:BSD_licence|BSD]]) license: Free speech as a pragmatic choice ====

'''BSD licenses''' are a family of [[wikipedia:Permissive_free_software_license|permissive free software license]]s. The original license was used for the [[wikipedia:Berkeley_Software_Distribution|Berkeley Software Distribution]] (BSD), a [[wikipedia:Unix_like|Unix-like]] operating system after which it is named.
The licenses have fewer restrictions on distribution compared to other free software licenses such as the [[wikipedia:Gnu_public_license|GNU General Public License]] or even the default restrictions provided by [[wikipedia:Copyright|copyright]], putting them relatively closer to the [[wikipedia:Public_domain|public domain]].

In distinction to viral license, BSD licenses are open source, that is, they allow inspection of source code in order to improve unlimited redistribution. Yet, there is no obligation of distribution under copyleft terms. One of the initial BSD licenses is shown below.

{{Infobox software license
 | name            = New BSD License
 | author          = [[wikipedia:Regents_of_the University_of_California|Regents of the University of California]]
 | copyright       = [[wikipedia:Public_Domain|Public Domain]]
 | released            = 1999-07-22 &lt;ref name=&quot;update&quot;&gt;{{cite web
   |url=ftp://ftp.cs.berkeley.edu/pub/4bsd/README.Impt.License.Change
   |title=To All Licensees, Distributors of Any Version of BSD
   |publisher=University of California, Berkeley
   |date=1999-07-22
   |accessdate=2006-11-15
  }}&lt;/ref&gt;
 | OSI approved    = Yes
 | Free Software   = Yes
 | GPL compatible  = Yes
 | copyleft        = No
}}


&lt;pre&gt;
Copyright (c) &lt;year&gt;, &lt;copyright holder&gt;
All rights reserved.

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:
    * Redistributions of source code must retain the above copyright
      notice, this list of conditions and the following disclaimer.
    * Redistributions in binary form must reproduce the above copyright
      notice, this list of conditions and the following disclaimer in the
      documentation and/or other materials provided with the distribution.
    * Neither the name of the &lt;organization&gt; nor the
      names of its contributors may be used to endorse or promote products
      derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS &quot;AS IS&quot; AND
ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL &lt;COPYRIGHT HOLDER&gt; BE LIABLE FOR ANY
DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
&lt;/pre&gt;

The '''permissive nature''' of the BSD license allows '''companies''' to distribute derived products as proprietary software without exposing source code and sometimes intellectual property to competitors. Searching for strings containing &quot;University of California, Berkeley&quot; in the documentation of products, in the static data sections of binaries and ROMs, or as part of other information about a software program, will often show BSD code has been used. This permissiveness also makes BSD code suitable for use in '''open source products''', and the license is compatible with many other open source licenses. The permissive nature of the BSD license also allows derivative works of code released originally under the BSD license to become less permissive with time.

====  [[wikipedia:Image:GNU_Lesser_General_Public_License|GNU Lesser General Public License (LGPL)]] : A compromise between GPL and BSD ====


{{ infobox software license
| name            = GNU Lesser General Public License
| author          = [[Free Software Foundation]]
| version         = 3
| copyright       = Free Software Foundation, Inc.
| date            = June 29, 2007
| Debian approved = Yes
| Free Software   = Yes
| GPL compatible  = Yes
| copyleft        = Yes
| linking         = Yes
}}
The '''GNU Lesser General Public License''' (formerly the '''GNU Library General Public License''') or '''LGPL''' is a [[wikipedia:free_software_license|free software license]] published by the [[Free Software Foundation]] (FSF). It was designed as a compromise between the strong-[[wikipedia:Copyleft|copyleft]] [[wikipedia:GPL|'''GPL''']] and [[wikipedia:Permissive_free_software_licence|permissive]] licenses such as the [[wikipedia:BSD_license|BSD license]]s. The GNU Library General Public License (as the LGPL was originally named) was published in 1991, and was the version number 2 for parity with GPL version 2. The LGPL was revised in minor ways in the 2.1 point release, published in 1999, when it was renamed the GNU Lesser General Public License to reflect the FSF's position that not all libraries should use it.

The LGPL places [[wikipedia:Copyleft|copyleft]] restrictions on the program itself but does not apply these restrictions to other software that merely links with the program. There are, however, certain other restrictions on this software.

The LGPL is primarily used for computer science software libraries, although it is also used by some stand-alone applications, most notably [[wikipedia:Mozilla|Mozilla]] and [[wikipedia:OpenOffice.org|Openoffice.org]] and sometimes media as well.

&lt;br&gt;
&lt;br&gt;
&lt;br&gt;

=== Open content licenses ===


Open content involves any kind of creative work, or content, published under an [[wikipedia:Open_Content_License|open content license (OPL)]] that explicitly allows copying and/or modification of the information by anyone. Hence, it includes the right to modify the work. Open content therefore is an alternative paradigm to the use of [[wikipedia:Copyright|copyright]] that ultimately creates a monopoly over the knowledge. One of the largest open content project is [[Wikipedia:Wikipedia|Wikipedia]].
It is interesting to notice that the [[wikipedia:Kuleuven|K.U.Leuven]] runs a [https://wiki.associatie.kuleuven.be/ WikiServer] that provides WikiSpace for educational purposes. Yet, the open character is lost because one needs to log in using the [http://toledo.kuleuven.be/  Toledo Blackboard] (a digital learning platform) in order to get access.

=== Open access licenses ===
[[wikipedia:Open_access_(publishing)|Open access (OA)]] refers to unrestricted online access to (mainly) peer-reviewed scholarly journal articles. While in scholarly publishing it is common to keep an article's content intact and to associate it with a fixed author, open access (publishing) mostly offers access to the material and defines a set of use (or re-use) rights, included in [[wikipedia:Creative_Commons_Licenses|Creative Commons Licenses]]. These Licenses replace the model of  '''all rights reserved''' of [[wikipedia:Copyright|copyright]] with the more flexible model of '''some rights reserved'''. In other words, through a [[wikipedia:Creative_Commons_Licenses|Creative Commons License]] creators communicate which rights they reserve and which rights they waive for the benefit of recipients or other creators (in order to make the material open content). Different models for financing OA journals exist. Some charge publication fees (paid by authors or by their funding agencies or employers) and some do not. Some of the no-fee journals have institutional subsidies , where others do not.

An example of open access (publishing) is [[wikipedia:Open_Courseware|Open CourseWare (OCW)]]. This is a collection of high quality course materials in a virtual learning environment created by universities ([[wikipedia:University_of_California_Berkeley|Berkeley]], [[wikipedia:MIT|MIT]], [[wikipedia:Yale|Yale]], [[wikipedia:University_Of_Notre_Dame|Notre Dame]], [[wikipedia:Michigan|Michigan]], [[wikipedia:Delft|TU_Delft]],  [[wikipedia:Stanford|Stanford]], …) and shared freely with the world via the internet.
In addition to that, there are several open access journals. We mentioned earlier that the problem with this type of journals is that their impact factor, based on the number of citations to articles published during the two to five preceding years, is initially very low. However, there are some well-known examples of open access journals, which are mainly to be found in the medical sciences, such as: [http://www.biomedcentral.com/  BioMed Central (BMC)], [http://www.medknow.com/ MedKnow Publications], [ http://publicaccess.nih.gov/   National Institute of Health Public Access (NIH)] and  [http://www.plos.org/ Public Library of science (PloS)]. Many open access academic publication platforms (with participation of K.U.Leuven academics) can be found within a wide variety of scientific fields: the field of [http://www.joser.org/ Software Engineering for Robotics], the BioMedical and Scientific field, chemistry, Molecular Imaging, etc.

=== Open standards ===

[[wikipedia:Open_standards|Open standards]] are certain specifications of technologies that are publicly available. They are meant to facilitate interoperability and data exchange among different products or services and are intended for widespread adoption. Some examples of open standards are: [[wikipedia:OpenDocument|Open Document Format (ODF)]] used by [[wikipedia:OpenOffice|Open Office]], [[wikipedia:Portable_Network_Graphics|Portable Network Graphics (PNG)]] images, [[wikipedia:Portable_Document_Format |Portable Document Format (PDF/X)]] designed by  Adobe Systems.

Although current policies actually discourage knowledge sharing, and in fact encourage knowledge privatisation, our short overview of initiatives and methods should clarify that it is possible to ‘open up’ knowledge production and distribution within universities. Perhaps it is important to start validating and supporting these initiatives and methods – as part of the public role of the university, and in order to make real their mission statements.

== K.U.Leuven as an &quot;Open&quot; university ? ==

=== Current ''open'' projects at K.U.Leuven ===


====  Open source software projects ==== 
Multiple project in '''OPTEC''': K.U.Leuven Center of Excellence: Optimization in Engineering 
::*  [http://acadotoolkit.org/ Acado Toolkit] - A Toolkit for Automatic Control and Dynamic Optimization (open-source), Boris Houska and Hans Joachim Ferreau (for C++ and Matlab).
::*[http://www.kuleuven.be/optec/software/qpOASES qpOASES]- Parametric Quadratic Programming for MPC (open-source), Hans Joachim Ferreau.
::*[http://www.esat.kuleuven.be/sista/lssvmlab/ lssvmlab]- a Least Squares Support Vector Machines toolbox. 
::*[http://people.mech.kuleuven.be/~dversche/timeopt/timeopt.html timeopt]  - Time Optimal Robot Trajectory Planning (open-source), Diederik Verscheure.
::*[http://www.cs.kuleuven.be/~karlm/glas/ glas]- Generic Linear Algebra Software (open-source), Karl Meerbergen.
::*[http://www.kuleuven.be/optec/software/rdp rdp]- Robust Dynamic Programming (open-source), Moritz Diehl and Jakob Björnberg.
::*[http://www.liftopt.org liftopt]- Nonlinear Optimization (open-source), Jan Albersmeyer.
::*[http://www.orocos.org orocos]- Smarter control in robotics &amp;amp; automation! (open-source), Herman Bruyninckx and Peter Soetens.
::*[http://www.kuleuven.be/optec/software/SCPCVX SCPCVX] - an interface for Sequential Convex Programming Methods (SCP) using the CVX package in Matlab.
The '''Bioinformatics''' Research Group 
::* [http://homes.esat.kuleuven.be/~saerts/software/toucan.php Toucan-2] is a workbench for regulatory sequence analysis on metazoan genomes : comparative genomics, detection of significant transcription factor binding sites, and      detection of cis-regulatory modules (combinations of binding sites) in sets of coexpressed/coregulated genes.
The '''VISICS''', a part of the Centre for Processing Speech and Images (PSI) within the K.U.Leuven
::* [http://homes.esat.kuleuven.be/~visit3d/webservice/v2/  ARC 3D Webservice] is a Family of Web Tools for Remote 3D Reconstruction.

==== Open content projects ====
Faculty or Arts, Katholieke Universiteit Leuven
::* MultiCampus Open Educational Resources: the case of [http://www.eadtu.nl/oerhe/?c=home OER-HE]

Department of mechanical engineering: Wikipages designed within a course embedded control systems
::*[[wikibooks:Embedded_Control_Systems_Design|Embedded control design systems]]


=== Suggestions for improvement ===

# KPI adaptation 

#Active support of 'open' initiatives 

# Creation of an 'open' course content platform

==Notes and references==
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{{Reflist}}

[[Category:Open academia]]
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  <page>
    <title>Wisdom/Curriculum</title>
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        <username>Lbeaumont</username>
        <id>278565</id>
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      <text bytes="14462" sha1="syclvdawol6nznjv67q75mpjzbdummq" xml:space="preserve">==Applied Wisdom==
{{TOC right }}
[[w:T._S._Eliot|T. S. Eliot]] asked:

{{cquote |Where is the wisdom we have lost in knowledge?
Where is the knowledge we have lost in information?}}

This ''Applied Wisdom Curriculum'' is being designed by asking how we can best prepare ourselves to solve the great universal problems that prevent us from realizing and enjoying all that is most important in life. Knowledge has not been enough; we need the broad scope, human perspective, and good judgment of ''wisdom''.

Shih-Ying Yang writes: “In the last analysis, individual actualization of conceptions of wisdom in real life, and the positive impact of these wise decisions and actions, may be the vehicle of the advance of human civilizations.”&lt;ref&gt; Yang, Shih-Ying. 2001. “Conceptions of Wisdom Among Taiwanese Chinese.” ''Journal of Cross-Cultural Psychology'' 32(6), November:662-680.&lt;/ref&gt;

This curriculum is based on the simple premise: If folly brings us problems, then perhaps [[wisdom]] can bring us solutions. The goal of the curriculum is to help you develop a tough mind and a tender heart.
{{By|lbeaumont}}
'''Pursuit of [[w:Well-being|well-being]]''' is the unifying theme for these courses, [[w:eudaimonia|eudaimonia]].

The collection of [[Wise Affirmations|wise affirmations]] can help you live more wisely each day. The [[Wise Living Toolkit]] assembles various resources that can help you live wisely.

Please choose courses from this curriculum and study them in any order that suits your interests. The [[Living Wisely]] course calls on these courses in a particular sequence intended to allow each new course to build upon concepts learned from previous courses.  The currently available courses are listed below in that sequence. 
[[File:Origins_and_progression_of_wisdom.webp|thumb|Origins and Progression of Wisdom]]
* [[Wisdom for the ages]] - Practical advice for [[Living Wisely|living wisely]]
* The [[Virtues]] — Attaining intrinsically valuable character traits
* [[Social Skills]] — Building Relationships
* [[Earning Trust]] — Relying on Another
* [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves
* [[Practicing Dialogue]] — Thinking Together
* [[Clarifying values]] — What we find most important
* [[What Matters]] — Identifying what is truly most significant to you, your family, community, nation, and world.
* [[Sleep Soundly]] — Attaining essential rest and restoration
* [[Stoic joy]] — Seeking tranquility.
* Courses from the [[Clear_Thinking/Curriculum|Clear Thinking]] curriculum. — Become more accurate and consistent in thinking.
** [[Facing Facts]] — Embracing Reality
** [[Evaluating Evidence]] — Seeking Reality
***[[Media literacy|Media Literacy]] — Identifying reliable sources
** [[Seeking True Beliefs]] — Excellence in the Quest for Knowledge
** [[Exploring Worldviews]] — Challenging our deeply embedded assumptions
** [[Deductive Logic]] — Tools for evaluating consistency
** [[Recognizing Fallacies]] — Describing inconsistencies
** [[Thinking Scientifically]] — Reliable ways of knowing
** [[Knowing_How_You_Know|Knowing How You Know]] — Developing and applying your own Theory of Knowledge.
** [[Intellectual Honesty]] — Seeking Real Good Together
** [[Socratic Methods]] — Seeking real good by questioning beliefs
***[[Street Epistemology]] — Exploring the basis for belief
**[[Exploring Social Constructs]] — Constructing Reality
*[[Finding Common Ground]] — Aligning concepts with reality
* [[Natural Inclusion]] — Experiencing the world ''from'' nature.
* [[Beyond Theism]] — A real basis for hope
* [[Global Perspective]] — Applying our Wisdom to meet the Grand Challenges
* Courses from the [[Emotional Competency]] curriculum:
** [[Emotional Competency]] — Developing the essential social skills to recognize, interpret, and respond constructively to emotions in yourself and others.
** [[Studying Emotional Competency]] — a path for studying the emotional competency material
** [[Dignity]] — Improving our world by learning to preserve dignity for all people
** [[Recognizing Emotions]] — Know how you feel
**[[Forming beliefs]] — Evaluating what you accept as true
** [[Resolving Anger]] — Resolving an urgent plea for justice and action
** [[Resolving Dominance Contests]] — The classic show down
**[[Confronting Tyranny]] — Resisting abusive power
** [[Overcoming Hate]] — Learning acceptance
**[[Appraising Emotional Responses]] — Explaining Events
** [[What you can change and what you cannot]] — Gaining the wisdom to know the difference
** [[Attributing Blame]] — Analyzing Cause and Effect
**[[Coping with Ego]] — Confronting the prime mover
** [[Apologizing]] — Expressing remorse.
** [[Forgiving]] — Choosing to overcome your desire for revenge
** [[Foregoing Revenge]] — Deescalating conflict
**[[Communicating Power]] — Projecting power as we speak
** [[Earning Trust]] — Relying on Another
** [[Practicing Dialogue]] — Thinking Together
** [[Candor]] — Gaining Common Understanding
** [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased.
** [[Transcending Conflict]] — Resolving contradictory goals
** [[True Self|Unmasking the True Self]] — Exploring the stories we tell ourselves about ourselves
**[[Finding Equanimity]] — Calm throughout the storm
**[[Cherishing awe]] — Connecting with vastness
**[[Alleviating Loneliness]] — Reconnecting
**[[Creating Communities]] — Belonging
** [[Toward congruence]] — Attaining alignment and agreement
* [[Pursuing Collective Wisdom]] — Improving collaborative decision making.
* [[Grand Challenges]] — The great problems and opportunities facing humanity
* Courses from the [[Possibilities/Curriculum|Possibilities curriculum]]:
**[[Creating Possibilities]]—Navigating problem space
**[[Unleashing Creativity]] — Welcoming new and useful ideas
**[[Thinking Tools]] — Boosting Imagination
**[[Problem Finding]] — Discovering the ''real'' problem
**[[Solving Problems]]—Creating solutions
**[[Embracing Ambiguity]]—Keep thinking
**[[Transcending Conflict]]—Resolving contradictory goals
**[[Playing]] — Enjoyable Activity
** [[Envisioning Our Future]] — Describing your vision of our future.
**[[Coming Together]]—Becoming wiser together
** [[Evolving Governments]] — Unleashing collaboration
** [[Evolving Money]]—Exchanging goods and services
* [[Dignity]] — Improving our world by learning to preserve dignity for all people
* [[Wisdom]] — Choosing Humanity
* [[Assessing Human Rights]] — Essential protections for every person
* [[Moral Reasoning]] — Knowing what to do
* [[Living the Golden Rule]] — Treating others as you want to be treated
** [[Understanding the Golden Rule]] — Treat others only as you consent to being treated in the same situation.
* [[Practicing Dialogue]] — Thinking Together
* [[Understanding Fairness]] — Your interpretation of what is fair is likely to be arbitrary and biased.
* [[Transcending Conflict]] — Resolving contradictory goals
* [[Limits To Growth]] — Recognizing the earth is finite
* [[Envisioning Our Future]] — Describing your vision of our future.
** [[A Journey to GameB]] — Life as it could be
** [[Intentional Evolution]] — Choosing our future
** [[Level 5 Research Center]] — The Next Big Thing
** [[Wisdom Research|The Wisdom and the Future Research Center]] — How can we wisely create our future?
*[[Finding Courage]] — Value-based action despite temptation.
* [[Doing Good]] — Take real good action.
* A [[Quiet Mind]] — Controlling Discursive Thought; cultivating Pure Awareness
*[[Guided Meditations]] — A selection of guided meditation scripts you may wish to practice.
* [[Living Wisely]] — Enjoy seeking ''real good'' throughout your life.
* [[Natural Inclusion]] — Experiencing the world ''from'' nature.

==Related Lectures and Essays==
Several of the courses in this applied wisdom curriculum include lectures or assign essays to read as part of the course work. Those lectures and essays are listed here, in alphabetical order.
* [[Living_Wisely/advance_no_falsehoods|Advance no Falsehoods]]
* [[Embracing Ambiguity/Ambiguity breeds schisms|Ambiguity breeds schisms]]
* [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]]
*[[Virtues/Humility/Authentic_Humility|Authentic Humility]]
*[[Virtues/Humility/Being 99.9% Ignorant|Being 99.9% Ignorant]]
*[[Assessing Human Rights/Beyond Olympic Gold|Beyond Olympic Gold]]
*[[Knowing How You Know/gallery/Choosing my beliefs|Choosing my beliefs]]
*[[Limits To Growth/Coping with Abundance|Coping with Abundance]]
* [[Knowing_How_You_Know/Divided_by_epistemology|Divided by epistemology]]
* [[Living Wisely/Does Seeking Real Good Transcend Metamodernism?|Does Seeking Real Good Transcend Metamodernism? ]]
* [[Finding Common Ground/Doubt and our Bayesian Brains|Doubt and our Bayesian Brains]]
*[[Limits To Growth/Earth at One Billion|Earth at One Billion]]
*[[Living_Wisely/Economic_Faults|Economic Faults]]
*[[Understanding_Fairness/fair_enough|Fair Enough]]
*[[Knowing How You Know/Friendly Persuasion|Friendly Persuasion]]
*[[Practicing_Dialogue/From_Demagoguery_to_Dialogue|From Demagoguery to Dialogue]]
*[[Living Wisely/Genesis of Debt|Genesis of Debt]]
*[[Evolving Governments/Good Government|Good Government]]
*[[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]]
*[[Virtues/How can you change another person?|How can you change another person?]]
* [[Understanding_Fairness/Luck,_Land,_and_Legacy|Luck, Land, and Legacy]]
*[[Knowing_How_You_Know/One_World|One World]]
*[[Facing_Facts/Perceptions_are_Personal|Perceptions are Personal]]
*[[Wisdom Research/Pinnacles|Pinnacles]]
*[[Living Wisely/Real, Good Insights|Real, Good Insights]]
*[[Facing Facts/Reality is our common ground|Reality is our common ground]]
* [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]]
*[[Beyond Theism/Resolving a Vital Paradox|Resolving a Vital Paradox]]
*[[Seeking_True_Beliefs/Science_is_like_a_living_tree|Science is like a living tree]]
*[[Living Wisely/Seeking Real Good|Seeking Real Good]]
*[[Problem_Finding/significance|Significance]] 
*[[Limits To Growth/Simply Priceless|Simply Priceless ]]
*[[Virtues/Spontaneous Conflict and Deliberate Restraint|Spontaneous Conflict and Deliberate Restraint]]
*[[Confronting Tyranny/The Hearing|The Hearing]]
*[[Thinking Scientifically/The role and limitations of scientific reduction|The role and limitations of scientific reduction]]
*[[Envisioning Our Future/The World We Want in 2075|The World We Want in 2075]]
*[[Global Perspective/tobacco road|Tobacco Road]]
*[[Global Perspective/Toward a Global Perspective—seeing through illusion|Toward a Global Perspective—seeing through illusion]]
* [[Envisioning_Our_Future/Toward_Compassion|Toward Compassion]]—Unleashing the power of kindness
*[[Beyond Theism/Transcending Dogma|Transcending Dogma]]
*[[Knowing How You Know/Tyranny of Evidence|Tyranny of Evidence]]
*[[Exploring_Worldviews/What_Fish_Don’t_See|What Fish Don’t See]]
* [[Beyond Theism/What there is|What there is]]
* [[Wisdom/Wisdom, Intelligence, and Artificial Intelligence|Wisdom, Intelligence, and Artificial Intelligence]]

== Research Projects ==
Several research projects are associated with this Applied Wisdom curriculum. These research projects include:
* [[Wisdom Research|The wisdom and the future research center]]
**[[Grand challenges/Causes of Suboptimal Life Experiences|Causes of Suboptimal Life Experiences]]
** [[Living Wisely/Improving our Social Operating Systems|Improving our Social Operating Systems]]
*The [[Level 5 Research Center]] is helping to shape the next big thing.

==Proposed Courses yet to be Developed==
Related Courses, some still to be developed, include:
* Determining ''What is''
** [[Evaluating Evidence|Evidence]]
*** [[Knowing How You Know/Tyranny of Evidence|The Tyranny of Evidence]]
** [[Introduction_to_logic|Logic and logical fallacies]]
** [[Knowing How You Know#What is a Theory of Knowledge?|Theory of Knowledge]] This is now available as the course [[Knowing_How_You_Know|Knowing How You Know]]. This course covers many of the topics listed above.
** [[Street Epistemology]] Learning to conduct genuine conversations that examine the foundations of belief.
** Using the metric system
*[[Living Wisely/Seeking Real Good|Seeking Real Good]]
** [[Thinking Scientifically|Scientific Method]]
** [[Beyond Theism/What there is|Taxonomy of Reality]]
** [https://www.goodreads.com/review/show/1455117057 Deep Pragmatism] 
** [[Moral Reasoning|Ethics]]
** Knowing what to do -- Getting from ''is'' to ''ought''.
* [http://www.wisdompage.com/2016%20Articles/Empathy%20and%20Wisdom%20Moss.pdf Developing Accurate Empathy] -- Why are they feeling that way?
* Systems Analysis
* Systems Design
** [[Problem Finding|Problem Seeking]]
* Rational Decision making
** The [[w:Analytic_Hierarchy_Process|Analytic Hierarchy Process]]
** Using [[w:Decision matrix|decision matrices]]
** [[w:Quality_function_deployment|Quality Function Deployment]]
** Choosing Excellence! 
* [[Critical_Thinking_Skills|Critical thinking]]
* [[w:Root_cause_analysis|Root cause analysis]]
* Understanding [[w:Risk|Risk]] — Estimating likelihood and consequence.
* [[Problem_solving|Problem solving]]
* [[Creativity]]
* [[w:Big_History|Big History]] — An integrated history of the universe from the Big Bang to the present
* [[Emotional Competency]]
* Marriage Excellence 
* [[Exploring Social Constructs|The nature of social constructs]]
* Designing social constructs for greater well-being
** Debugging Social Constructs
* Money Architectures — exploring implications and alternatives to national fiat currencies.
* Collective Wisdom — This is now available as the course [[Pursuing Collective Wisdom]].
*Forecasting using [[w:Bayes_theorem|Bayes Theorem]] 
* Inner Growth
* The wisdom of [[w:Ubuntu_(philosophy)|ubuntu]].
* [[w:Biomimicry|Biomimicry]] and sustainable design.
* Effecting change - How change propagates, or fails to propagate, through an organization or society.
** Influencing beliefs

You can help by becoming a student, improving the above list, or by developing one of these courses.

==References==
&lt;references/&gt;

[[Category:Applied Wisdom]]
[[Category:Curriculum]]</text>
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    <title>Haskell programming in plain view</title>
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      <timestamp>2024-12-16T02:07:01Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Lambda Calculus */</comment>
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==Introduction==
* Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]])
* Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]])
* Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]])
* Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]])
* Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]])
&lt;/br&gt;

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

&lt;/br&gt;

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

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


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

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

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

==Lambda Calculus==
* Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]])
* Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]])
* Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]])
* Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]])
* Encoding Datatypes 
:- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]])
:- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]])
:- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]])
:- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]])
:- Recursions ([[Media:LCal.9A.Recursion.20241216.pdf |pdf]])
&lt;/br&gt;
&lt;/br&gt;

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

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

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

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

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

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

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

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

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

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

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

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

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

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


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

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


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    <title>Clear Thinking/Curriculum</title>
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      <contributor>
        <username>Lbeaumont</username>
        <id>278565</id>
      </contributor>
      <comment>Added Evaluating Journalism Standards</comment>
      <origin>2692040</origin>
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      <text bytes="4811" sha1="qsejuao70po25036ti6t86p53wub085" xml:space="preserve">==Clear Thinking==
[[File:The Thinker, Rodin.jpg|thumb|upright=1.2|Think clearly.]]
This clear thinking curriculum is being developed as part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]] to help people become more accurate and consistent in their thinking.

An [https://rebelwisdom.substack.com/p/the-sensemaking-companion-section alternative curriculum], provided by Rebel Wisdom, is also available. 

Reading this essay on the [[Knowing How You Know/Height of the Eiffel Tower|Height of the Eiffel Tower]] can provide a good starting point for these studies.

{{By|lbeaumont}}
These courses are available now and establish the core of the curriculum:
* [[Facing Facts]] — Embracing Reality
* [[Evaluating Evidence]] — Seeking Reality 
* [[Seeking True Beliefs]] — Excellence in the Quest for Knowledge
* [[Exploring Worldviews]] — Challenging our deeply embedded assumptions
* [[Deductive Logic]] — Evaluating Consistency
* [[Recognizing Fallacies]] — Describing inconsistencies
* [[Knowing How You Know]] — Developing and applying your own Theory of Knowledge.
* [[Thinking Scientifically]] — Reliable ways of knowing
** [[Beyond_Theism/What_there_is|What There Is]] — Know what is real
* [[Intellectual Honesty]] — Seeking Real Good Together
* [[Socratic Methods]] — Seeking real good by questioning beliefs
*[[Exploring Social Constructs]] — Constructing Reality
*[[Finding Common Ground]] — Aligning concepts with reality
*[[Evaluating Journalism Standards]] — Seeking reliable information sources
*[[Resolving Cognitive Dissonance]]—Understanding, Identifying, and Addressing Inner Conflict
*[[Real Good Religion]]—Rebooting spiritual practice
*[[Understanding Emergence]]—Exploring the possible
* [[Natural Inclusion]] — Experiencing the world ''from'' nature.

Courses in this curriculum address [[w:Common_Core_State_Standards_Initiative|common core standards]] requirements for:  [http://www.corestandards.org/Math/Practice/MP3/ Standards for Mathematical Practice » Construct viable arguments and critique the reasoning of others].

The following courses are proposed but yet to be developed:

* Forecasting with Bayes’ Theorem — Integrating Evidence
* Clarifying Probabilities — Using Natural Frequencies
* Understanding Risk — Exposure to loss
* [[w:Inductive_reasoning|Inductive Reasoning]] — Evaluating expectations
** The [[w:Problem_of_induction|problem of induction]]
* [[w:List_of_cognitive_biases|Cognitive Errors and Distortions]] — Circumventing Occlusions
* [[w:Rhetoric|Rhetoric]] — Appraising the allure
* [[Recognizing_Fallacies#Taming_Wild_Fallacies|Analyzing Real Arguments]] — Evaluating rhetoric

Until the courses in this curriculum are complete, students may benefit from the courses now available in the Khan Academy [https://www.khanacademy.org/partner-content/wi-phi/critical-thinking Critical Thinking curriculum].

==References==
The following references are likely to be used in developing courses in this curriculum.
* [http://plato.stanford.edu/entries/logic-informal/ Informal Logic], Stanford Encyclopedia of Philosophy
* [http://plato.stanford.edu/entries/logic-classical/ Classical Logic], Stanford Encyclopedia of Philosophy
* [https://open.bccampus.ca/find-open-textbooks/?uuid=5d41a649-ce0f-4462-bc3d-564568b5c857&amp;contributor=&amp;keyword=&amp;subject= ''Introduction to Logic and Critical Thinking''], by Matthew J. Van Cleave
* {{cite book |last=Dobelli |first=Rolf |date=May 6, 2014 |title=The Art of Thinking Clearly  |publisher=Harper Paperbacks |pages=384 |isbn=978-0062219695}}
* [https://betterhumans.coach.me/cognitive-bias-cheat-sheet-55a472476b18#.rs9aqmhd3 Cognitive bias cheat sheet], Because thinking is hard. Buster Benson
* More on [http://bigthink.com/robby-berman/a-chart-of-brain-busting-cognitive-biases-hang-it-on-your-wall cognitive biases]. 
* {{cite book |last=Fisher |first=Alec |date= |title=The Logic of Real Arguments |publisher=Cambridge University |pages=234 |isbn=978-0521654814}}
* [https://medium.com/@yegg/mental-models-i-find-repeatedly-useful-936f1cc405d#.azrwzjjbu Mental Models I Find Repeatedly Useful], Gabriel Weinberg, July 5, 2016
* {{cite book |last=Dennett |first=Daniel C. | author-link=w:Daniel_Dennett |date=May 5, 2014 |title=Intuition Pumps And Other Tools for Thinking |publisher=W. W. Norton &amp; Company |pages=512 |isbn=978-0393348781}}
*{{cite book |last=Wilson |first=Edward Osborne |author-link=w:E._O._Wilson |date=Mar 30, 1999 |title=[[w:Consilience_(book)|Consilience: The Unity of Knowledge]] |publisher=Vintage |pages=384 |isbn=978-0679768678}}
* [https://www.criticalthinking.org/store/products/core-set-of-critical-thinkers-guides/559 Core set of Critical Thinker's Guides].


[[Category:Curriculum]] [[Category:Applied Wisdom]]
{{Clear Thinking}}</text>
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  <page>
    <title>Python programming in plain view</title>
    <ns>0</ns>
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    <revision>
      <id>2692140</id>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Using Libraries */</comment>
      <origin>2692140</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2647" sha1="6l8qh1haketz1ygk7qudz5zq0cefb69" xml:space="preserve">==''' Part I '''==
&lt;!----------------------------------------------------------------------&gt;
===  Introduction ===

* Overview 
* Memory 
* Number 

&lt;!----------------------------------------------------------------------&gt;


=== Python for C programmers ===

* Hello, World! ([[Media:CProg.Hello.1A.20230406.pdf |pdf]])
* Statement Level  ([[Media:CProg.Statement.1A.20230509.pdf |pdf]])
* Output with print
* Formatted output
* File IO

&lt;!----------------------------------------------------------------------&gt;


=== Using Libraries ===

* Scripts ([[Media:Python.Work2.Script.1A.20231129.pdf |pdf]])
* Modules ([[Media:Python.Work2.Module.1A.20231216.pdf |pdf]])
* Packages ([[Media:Python.Work2.Package.1A.20241207.pdf |pdf]])
* Libraries ([[Media:Python.Work2.Library.1A.20241216.pdf |pdf]])
* Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

===   Handling Repetition ===
 
* Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]])

* Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling a Big Work ===

* Functions ([[Media:Python.Work1.Function.1A.20230529.pdf |pdf]])
* Lambda ([[Media:Python.Work2.Lambda.1A.20230705.pdf |pdf]])
* Type Annotations ([[Media:Python.Work2.AtypeAnnot.1A.20230817.pdf |pdf]])

  
&lt;!----------------------------------------------------------------------&gt;

=== Handling Series of Data ===

* Arrays ([[Media:Python.Series1.Array.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series2.Tuple.1A.pdf |pdf]])
* Lists ([[Media:Python.Series3.List.1A.pdf |pdf]])
* Tuples ([[Media:Python.Series4.Tuple.1A.pdf |pdf]])
* Sets ([[Media:Python.Series5.Set.1A.pdf |pdf]])
* Dictionary ([[Media:Python.Series6.Dictionary.1A.pdf |pdf]])

&lt;!----------------------------------------------------------------------&gt;

=== Handling Various Kinds of Data ===
  
* Types 
* Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]])
* Files ([[Media:Python.Data4.File.1.A.pdf |pdf]])


&lt;!----------------------------------------------------------------------&gt;

=== Class and Objects ===
  
* Classes &amp; Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]])
* Inheritance

 
&lt;!----------------------------------------------------------------------&gt;
&lt;/br&gt;

== Python in Numerical Analysis  ==
 
&lt;/br&gt;
&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]

==External links==
* [http://www.southampton.ac.uk/~fangohr/training/python/pdfs/Python-for-Computational-Science-and-Engineering.pdf Python and Computational Science and Engineering]</text>
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  <page>
    <title>Engineering Projects/Engineering Drawing</title>
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      <text bytes="2421" sha1="tnhb03s2dcwb979bwaje97bvjs4886v" xml:space="preserve">[[File:Comparison_of_graphical_projections.svg|thumb|Graphical projections]]&lt;nowiki&gt;{{&lt;/nowiki&gt;[[Template:Soft redirect|Soft redirect]]&lt;nowiki&gt;|Blueprint Drawing }}&lt;/nowiki&gt;
*''See also [[w:Category:Graphical projections|'''Category:Graphical projections''']] on Wikipedia''

Engineering drawing is primarily used to provide information about an object rather than a pleasing image. It must convey enough information to create or use the object in the physical world.

Engineering drawings take many forms, and usually represent a three dimensional object. They tend to avoid color, instead using black lines only. Lines can be of varying styles and thickness to indicate outlines, dimensions, hidden detail, center lines, cross hatching and projections. Assembly drawings can show how a number of components can fit together, and it may take several drawings to fully describe a complex item such as a building, furnishing, design item, mechanism, or machine.

Broadly, technical drawings can be grouped as pictorial (less common) or orthographic. Pictorial drawing techniques provide a visual impression of what something looks like. These include:
* Isometric
* Planometric
* Oblique
* Perspective

[insert pictorial example(s)]

Orthographic drawings on the other hand present the information as a group of two dimensional representations. Each of these provides an outline and details looking from a particular direction. Orthographic drawings are like tracings of the outlines and key features of a photograph, and images are typically taken from three mutually perpendicular (orthogonal) directions. This translates conveniently into the physical world where someone needs to make or use the object. Where the object being described is complex, auxiliary views may also be provided showing the object from a suitable direction.

[insert engineering drawing example]

The actual views provided depend on the purpose of the drawing. Some special cases include house plans, which may only include a single view from the top to indicate the layout of rooms including walls, doors, windows and fixtures. A two dimensional sign might also only require a single view.

[insert building layout example]  

(potential headings)

Angles of projection
First Angle projection
Third Angle projection
Terms
Lines
Dimensions
Scale
Title blocks
Sketches
Physical drawings
Computer Aided Design (CAD)

[[Category:Engineering data]]</text>
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      <text bytes="2324" sha1="sjfj2y88jzsfh00qud9x1ek1lisgxvk" xml:space="preserve">[[File:Comparison_of_graphical_projections.svg|thumb|Graphical projections]]
*''See also [[w:Category:Graphical projections|'''Category:Graphical projections''']] on Wikipedia''

Engineering drawing is primarily used to provide information about an object rather than a pleasing image. It must convey enough information to create or use the object in the physical world.

Engineering drawings take many forms, and usually represent a three dimensional object. They tend to avoid color, instead using black lines only. Lines can be of varying styles and thickness to indicate outlines, dimensions, hidden detail, center lines, cross hatching and projections. Assembly drawings can show how a number of components can fit together, and it may take several drawings to fully describe a complex item such as a building, furnishing, design item, mechanism, or machine.

Broadly, technical drawings can be grouped as pictorial (less common) or orthographic. Pictorial drawing techniques provide a visual impression of what something looks like. These include:
* Isometric
* Planometric
* Oblique
* Perspective

[insert pictorial example(s)]

Orthographic drawings on the other hand present the information as a group of two dimensional representations. Each of these provides an outline and details looking from a particular direction. Orthographic drawings are like tracings of the outlines and key features of a photograph, and images are typically taken from three mutually perpendicular (orthogonal) directions. This translates conveniently into the physical world where someone needs to make or use the object. Where the object being described is complex, auxiliary views may also be provided showing the object from a suitable direction.

[insert engineering drawing example]

The actual views provided depend on the purpose of the drawing. Some special cases include house plans, which may only include a single view from the top to indicate the layout of rooms including walls, doors, windows and fixtures. A two dimensional sign might also only require a single view.

[insert building layout example]  

(potential headings)

Angles of projection
First Angle projection
Third Angle projection
Terms
Lines
Dimensions
Scale
Title blocks
Sketches
Physical drawings
Computer Aided Design (CAD)

[[Category:Engineering data]]</text>
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  <page>
    <title>Open Educational Resources/geo-located OER</title>
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      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Learning Tasks */</comment>
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      <text bytes="13737" sha1="9vq4ifoniuxdsypd9uad0fxbzw5bptk" xml:space="preserve">Even if learning resources are presented at Wikiversity as a digital media, learning processes itself taking place with interaction with your 
* social,
* cultural,
* educational,
* scientific,
* ...
environment. Taking these requirements and constraints into account leads to geo-located Open Educational Resources that are adapted to the location where they are used.

== Attach Learning Resouces to Geolocations ==
'''OER attached to Geolocations:''' Imagine your could attach digital Open Educational Resource to a real geolocation. The geolocation is e.g. a small river and use ''&quot;[[Augmented Reality Tutorial]]&quot;'' for visualization of OER:

* Attach a video that shows the place from a bird view.
* Provide a online questionnaire about the biological environmental observations at the river.
* Attach an image, how the river will look in winter.
* Visualize the work of mircro-bacteria doing work in the river.
* Show historic images of the river from the year 1912 and let the visitors analyse the human alteration in the river bed.

== Wiki Products and Geolocated OER ==
If we analyse producte of the WikiMedia foundation for geolocation attributes, the following can be identified:
* '''(Language and Geolocation)''' language of a learning resource in Wikiversity can be to make a rough af´sssignment to a geographical area in which the language is spoken.
* '''[[Wikipedia:Latitude|Latitude]]/[[Wikipedia:Longitude|Longitude]]''' can make a reference to a [[w:Point of interest|Point of Interest (POI)]] or a building (see coordinates [[Wikipedia:Vienna International Centre|Vienna International Centre]] upper right corner of article&lt;ref&gt;Nielsen, F. A. (2016, April). Literature, Geolocation and Wikidata. In Wiki@ ICWSM.&lt;/ref&gt;)
* '''(Geonames)''' Denomination of a country, city, address, ...
* '''(Geographical Area)''' define an area on the globe by a sequence of several geolocations (points). Let &lt;math&gt;P:=(x,y)&lt;/math&gt; be a point as a single longitude/latitude pair &lt;math&gt;(x,y)&lt;/math&gt;. A &quot;triangle&quot; is a geographical area consisting of three points :&lt;math&gt;\langle P_1,P_2,P_3 \rangle = \langle (x_1 , y_1),(x_2 , y_2),(x_3 , y_3) \rangle&lt;/math&gt;.

== Learning Tasks ==
[[File:Foro Romano Musei Capitolini Roma.jpg|thumb|Forum Romanum - Rome - Archeological Site]]
* '''(Geo-located AR)''' [https://github.com/jeromeetienne Jerome Etienne] create AR.js based on the libraries Aframe and [https://threejs.org/ three.js library]. Nicolo Carpignoli worked on Geolocated Augmented Reality and merge the concept with a marker based augmented reality (see [[3D Modelling]]). In 2020 the Open Source Augemented Reality package AR.js contains 3 different AR approaches:
:* Marker based (initiated by Jerome Etienne)
:* Geo-location based (developed by Nicolo Carpignoli)
:* Image Tracking based (AR.js Community)
:In this context we can geolocated Open Educational Resources 
:* with QR-Codes at special places that encode small infomation as content directly or a reference/URL to OER that could be digital media of any kind that supports learner at the geolocation she/he is currently at.
:* Augemented Reality visualizing additional content as specific geolocations (see also [https://ar-js-org.github.io/AR.js-Docs/#what-web-ar-means-augmented-reality-on-the-web animated examples of AR.js on project web portal])
* '''([[Photogrammetry]])''' Explain how [[Photogrammetry]] or historical site (e.g. [[w:en:Forum Romanum|Forum Romanum in Rome]]) can be used together with [[3D Modelling/Examples/AR with Markers|3D Modelling on Markers]]
* '''([[Wikiversity:Edit-a-thon]])''' visit a specific location with a group of educators, teachers and identify topics for an edit-a-thon for this specific geo-location. Run the edit-a-thon and create Open Educational Resources in Wikiversity with the educators. Perform a follow-up meeting for the edit-a-thon with the edit-a-thon participants and test the geo-location OER with the group of learners, students and collect their feedback for improving the learning resource.

== History of Approach ==
Community of Practice in the [http://at6fui.weebly.com/ Action Team 6 Follow Up initiative] and international approaches of [[Risk Management|Risk Mitigation strategies]] analysed the peaceful use of space technology (see [[w:United_Nations_Committee_on_the_Peaceful_Uses_of_Outer_Space|UN-COPUOS]]&lt;ref&gt;Hosenball, S. N. (1979). The United Nations Committee on the peaceful uses of outer space: Past accomplishments and future challenges. J. Space L., 7, 95.&lt;/ref&gt;). Having risk mitigation strategies for communities in mind, capacity building and learning is one of the key elements, that help organisations to support the communities exposed to risks. United Nations support that in many areas (Climate Change, Humantarian Aid, Disaster Management, Global Health). This learning resource is based on results of the community of practice in AT6FUI/EFG-SGH by piloting protoyping ideas (see [http://at6fui.weebly.com AT6FUI/Expert Focus Group for Space and Global Health]. But the learning resource is designed in way that generic princples like &quot;attaching digital information to geolocation&quot; are used as topic of the learning resource, because the generic geospatial principles can be used and transfered to other educational settings or use-cases for capacity building as well.

== Geolocation OER for Risk Management == 
To learn about strategies to mitigate the risk includes improvement of [[Risk Literacy|risk literacy]]. The '''access to the open educational resources for everyone''' is necessary to reach people that are exposed to risks. [[w:Wikiversity|Wikiversity]] provides OER and it at the same time a collaborative environment for teachers and learner. At the interface of organisational capacity building the concept of an ''One fits all'' approache does not work in general and especially not for educational resources. Language, social and cultural determinants are relevant how learner perceive risks. This implies that learning resource had to be tailored to this determinants. Wikiversity allows the alteration, remix, adaptation to local and regional requirements and constraints. So geospatial aspects are relevant for educational resources as well. Wiki products allow the attachment of articles to specific geolocation (see [[w:Vienna_International_Centre|United Nations Building VIC]] in Vienna - upper right corner). The georeferencing can be used by service like [http://www.geonames.org/ http://www.geonames.org/]&lt;ref&gt;Speiser, S., &amp; Harth, A. (2010, November). Towards linked data services. In Proceedings of the 2010 International Conference on Posters &amp; Demonstrations Track-Volume 658 (pp. 157-160). CEUR-WS. org.&lt;/ref&gt;. Based on such a service the latitude and longitude of the learber can be used find nearby learning resources in Wikiversity. Even augemented reality tools like [http://www.mixare.org/ Mixare] can used along with the webservices to place geolocated learning resources digital information that make reference to geolocation where the user is. 

=== Design Example of geolocated OER ===
Assume a group of learner will visit the [[w:Niagara Falls|Niagara Falls]]. Teacher could enrich the geolocation with additional media:
:* video from a helicopter,
:* looking into the past: the first snapshots of the location from beginning of photography,
:* Open Literature, open poems, that refer to the geolocation,
:* mircoscope views about the aquativ life,
:* how will the Niagara Falls evolve in next 100000 year as projections in the future
:* how does the place look in Winter, Spring, Summer, Autumn, ... temporal aspect
:* ...
The geolocated OER involve learning tasks that trigger the interaction of the learner with the environment.

=== Images Maps - A way of visual exploration of Learning Resources ===
&lt;imagemap&gt;File:Annweiler Queich 05.JPG|Click on Water Wheel - Image River: Queich, City: Annweiler, Country: Germany|350px|thumb

rect 1178 344 1508 1243 [[Wikipedia:Water_wheel|Water Wheel Link to Wikipedia]]
&lt;/imagemap&gt;

An image is a standard way for illustration of text documents. A image map attached links to certain areas on the image. Learners can click on the image and explore details and further information. A geolocated Learning Resoure can use this concept to allow exploration of a geolocation without being at this geographic location. This is helpful as a preparation for a physical visit of that place (see [[Risk_Literacy/Real_World_Labs|Real World Lab]]). The image on the right is a small example of on image map.

== Commercial Use of tailored information for geolocation ==
Geolocted OER applied a basic geospatial principle to OER, which is widely used in the commercial sector:
* Games like P0K0M0N G0 allows guiding users to commercially interesting location, 
* advertisments from mobile phone users can be placed the ads tailored to geo-location of mobile phone user (see [[Commercial Data Harvesting]]), 
* travel guides can provide digital information for the geolocation/city/leisure park on mobile devices, 
* ... 
All applications show the same generic geo-spatial principle:
: Provide information tailored to the geo-location of the user

== Extend Perspective of Teachers on the Design of Learning Resources ==
The use of digital devices has changed and in development countries the era of desktop computers skipped and the first computer will be mobile device. Have the geospatial principle in mind the major shift can be summarized as: 
: ''Instead of sitting in front of Desktop Computer with internet connection watching multimedia content about a geolocation, the mobile device learners at the geolocation interact with the physical environment around with the opportunity to enrich the experience with digital support and context information about the location they are currently living and working in.''  
The awareness of teachers about this basic geo-spatial principle changes the design of learning resources with a combination of learning resources used in the classroom and outside the classroom where the learners can get real experiences with the subject considered in the learning process. 

'''(Teachers Training)''' Learning resources for teachers training in Wikiversity start with 
* general awarness about the application of geospatial principles and  
* then the learning task in teachers training is the application of the principle their own expertise and their country or region they live in. 
* finally a more advanced task is to decompose the developed learning resource in parts that are applicable for other geolocations and areas as well and into parts that just refer to the specific geolocation.

== Importance of geolocated OER for Risk Management ==
If someone lives in the [[w:Antarctica research station|Antarctica research station]] then it doesn't make sense to inform the researcher about malaria risks. If you're going on holiday to the Kruger National Park, then risk mitigation information is helpful about the use of mosquito repellent, Mosquito nets or take malaria prophylaxis. 

== App Development for geospatial OER in Wikiversity==
[http://www.mixare.org/static/index.html Mixare] is a good starting point to explore the possibilities of such an approach. Also the software design analysis is helpful to learn about the import of geospatial &quot;nearby&quot;-databases with OpenSource technology. 

Looking teachers without IT background a productive use of these approach has a requirement, that with assignment of a geolocation in a Wikipedia learning resource, this learning resource or any other wikipedia article should be available in the Augmented Wiki Browser. No additional workload should be necessary for teacher to import the &quot;nearby&quot;-learning resource of &quot;nearby&quot;-Wikipedia article.  

From the software development side the Mixare-approach needs the refactoring of code or a reimplementation as multiplatform application as HTML5/JS/CSS app (suggestion might be moved to a meta area of Wiki products in general). 

Crowd funding could support the development of a Open Source app that allows teachers 
* to use geolocated learning resources, 
* place learning resource as teachers to specific geolocation and 
* have something an Open Source MediaWiki Browser structural equivalent to [https://www.mixare.org Mixare] that allows teacher to create geolocated OER just by placement of latitude and longitude in the OER.
A major improvement of Mixare is necessary mainly for the reduction the workload for teachers and not from the usability for learners to retrieve Open Educational Resource that a relevant to their CURRENT geolocation. Technically an improvement of access to mobile phone sensors especially GPS location must be tested for current devices and operating systems (stability)

==Think globally, act locally==
The approach &quot;''Think globally, act locally''&quot;&lt;ref&gt;Kefalas, A. G. (1998). Think globally, act locally. Thunderbird International Business Review, 40(6), 547-562.&lt;/ref&gt; can be applied to learning resources. Global approaches in a learning resouce lead to local or regional implementations/specialisation of OER that respects local cultural, social requirements and the local constraints to selected location as learning resource.

== Sea also ==
* '''[[Risk_Literacy/Real_World_Labs/web-based_exploration|Real World Lab]]:''' application of geolocated OER tailored for the Real World Lab
* '''[[3D Modelling]]''' as Open Educational Resources for a [[Digital Learning Environment]] or [[Capacity Building]]
* '''[[Space and Global Health/geo-located OER|geo-located OER in the context of Space and Global Health]]'''

== References ==

[[Category:Open educational resources]]
[[Category:Think globally - act locally]]</text>
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    <title>The necessities in Microprocessor Based System Design</title>
    <ns>0</ns>
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      <text bytes="5910" sha1="nm47kl7aad57r7a1hok8lqcwl5vfq2h" xml:space="preserve">== '''Background''' ==

'''Combinational and Sequential Circuits'''
* [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] 
* [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]]

'''FSM'''
* [[Media:DD3.A.3.FSM.20131030.pdf |FSM]]
* [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]]

'''Tiny CPU Example'''
* [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]]
* [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]]
* [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]]
* [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]]
&lt;/br&gt;

== '''Microprocessor Architecture''' ==
* ARM Architecture 
: - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]])
: - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]])
* ARM Organization 

* ARM Cortex-M Processor Architecture 
* ARM Processor Cores
&lt;/br&gt;

== '''Instruction Set Architecture''' ==
* ARM Instruction Set 
: - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]])
: - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]])
: - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]])
: - Assembler Format
::  - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]])
::  - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]])
::  - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]])
::  - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]])
: - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]])
* Thumb Instruction Set 

&lt;/br&gt;

== '''Assembly Programming''' == 


=== ARM Assembly Programming (I) ===
* 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]])
* 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]])
* 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]])
* 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20230726.pdf |pdf]])
* 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]])
* 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) 
* 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) 
* 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) 
* 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]])  
* 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) 
* 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) 
* 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) 
::  

::

=== ARM Assembly Programming (II) ===
::  
* 1. Branch and Return Methods ([[Media:ARM.2ASM.Branch.20241216.pdf |pdf]]) 
* 2. PC Relative Addressing ([[Media:ARM.2ASM.PCRelative.20241123.pdf |pdf]]) 
* 3. Thumb instruction Set ([[Media:ARM.2ASM.Thumb.20241123.pdf |pdf]]) 
* 4. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) 
* 5. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) 
* 6. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) 
* 7. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) 
* 8. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) 
* 9. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20230610.pdf |pdf]]) 
* 10. Tail Chaining ([[Media:ARM.2ASM.TailChain.20230816.pdf |pdf]]) 


&lt;/br&gt;
* ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]])
::

=== ARM Assembly Programming (III) ===
* 1. Fixed point arithmetic (integer division)
* 2. Floating point arithmetic 
* 3. Matrix multiply

=== ARM Linking ===
* arm link ([[Media:arm_link.20211208.pdf |pdf]]) 


&lt;/br&gt;

=== ARM Microcontroller Programming ===
* 1. Input / Output 
* 2. Serial / Parallel Port Interfacing
* 3. Analog I/O Interfacing
* 4. Communication


&lt;/br&gt;


== '''Memory Architecture''' == 
&lt;/br&gt;

=== '''Memory Hierarchy''' === 
&lt;/br&gt;

=== '''System and Peripheral Buses''' === 
&lt;/br&gt;

=== '''Architectural Support''' === 
* High Level Languages
* System Development
* Operating Systems
&lt;/br&gt;


== '''Peripheral Architecture''' == 
&lt;/br&gt;

=== '''Vectored Interrupt Controller ''' === 
&lt;/br&gt;

=== '''Timers ''' === 
* Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) 
* Real Time Clock
* Watchdog Timer
&lt;/br&gt;

=== '''Serial Bus''' === 
* '''UART''' : Universal Asynchronous Receiver/Transmitter  ([[Media:ARM.4ASM.UART.20220924.pdf |pdf]]) 
* '''I2C''' : Inter-Integrated Circuit 
* '''SPI''' : Serial Peripheral Interface 
* '''USB''' : Universal Serial Bus Device Controller
&lt;/br&gt;

=== '''I/Os ''' === 
* General Purpose Input/Output ports (GPIO)
* Pulse Width Modulator
* Analog-to-Digital Converter (ADC)
* Digital-to-Analog Converter (DAC)

&lt;/br&gt;


&lt;!-- == '''Interrupts and Exceptions ''' == --&gt;
&lt;/br&gt;

== '''Synchrnoization'''==
&lt;/br&gt;

=== H/W and S/W Synchronization === 
* busy wait synchronization
* handshake interface
&lt;/br&gt;

=== Interrupt Synchronization === 
* interrupt synchronization
* reentrant programming
* buffered IO
* periodic interrupt
* periodic polling 
&lt;/br&gt;

==''' Interfacing '''==
&lt;/br&gt;

=== Time Interfacing === 
* input capture
* output compare
&lt;/br&gt;


=== Serial Interfacing === 
* Programming UART
* Programming SPI
* Programming I2C
* Programming USB 
&lt;/br&gt;

=== Analog Interfacing === 
* OP Amp
* Filters
* ADC 
* DAC 
&lt;/br&gt;

== '''Old materials''' ==

=== '''Instruction Set Architecture''' ===
* ARM Instruction Set 
:: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]])
:: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]])
:: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]])
* Thumb Instruction Set 
* ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]])
* ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]])
&lt;/br&gt;


&lt;/br&gt; 
go to [ [[Electrical_%26_Computer_Engineering_Studies]] ]</text>
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  <page>
    <title>AP Psychology/Introduction</title>
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      <id>2692043</id>
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      <timestamp>2024-12-15T19:35:46Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Study Notes */ including the date</comment>
      <origin>2692043</origin>
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      <text bytes="5071" sha1="imtupxt0ikp8qqakkechn5y3ebktugd" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
Introduction to the AP Psychology course and the basic psychological principles needed for the remaining sections in this course.

== Study Notes ==
'''Psychology''': The scientific study of the mind and behavior&lt;br&gt;
'''Empiricism''': A concept from John Locke, in which knowledge in science should be backed up by experimentation and observation. 

The origin of psychology goes back to Ancient Greek, in which philosophers started questioning themselves and human nature. Psychology officially becomes its own science in 1879. 
;Mind's structure/function
*'''Structuralism''' - A school of psychology that teaches &quot;introspection&quot; (looking within) and looks into the mind interpreting subjects as structures (&quot;elements of the mind&quot;). ''Edward Titchener'' is the founder of this school of thought, in which he wanted to discover the immediate thoughts relating to a certain item (immediate feelings about fear: sweaty, fast heart-beat). It is proven to be unreliable as recollections differ between person to person and people are known to recount incorrectly. 
*'''Functionalism''' - A school of psychology that teaches about the performance/function of a certain subject (fear: awakens you, gets you out of danger, more careful, act faster). ''William James'' was inspired by an evolutionist, Charles Darwin, and sought to discover the ways these subjects evoked and allowed us to act. 
;People
[[File:Wilhelm Wundt.jpg|thumb|right|Wilhelm Wundt, the first recorded psychologist]]
'''Wilhem Wundt''' was the first psychologist to establish a psychology lab (in Germany) in 1879. He performed an experiment where:
# The ball is dropped, the lever is pushed.
# The ball is dropped, but the lever is only pushed if the person is ''consciously aware of the ball dropping''. 

'''G. Stanley Hall''' was the first president of the APA, first American to receive a degree in psychology and established the first US psychology lab. 

'''Mary Whiton Clarkins''': Denied a psychology degree because she's a woman. First female president of the APA.

'''Margaret F. Washburn''': Student of Edward Titchener. The first female to receive a psychology degree. Second female president of the APA and wrote ''The Animal Mind''. 

'''Dorthea Dix''': Wanted more humane treatment of the mentally ill.
===Psychology Branches===
====7 Views of Human Nature====
=====Psychodynamic=====
Also known as &quot;psychoanalytic&quot;, '''psychodynamic''' is the thought in which '''human behavior stems from unconscious desires and behavior from his/her early years (the past)'''. This includes repressed memories and dream interpretation. This thought process originates from ''Sigmund Freud''. For example, a psychologist with the psychodynamic approach might view an outburst as originating from unconscious anger.

===== Cognitive =====
In the cognitive approach, the main target is the way a person '''thinks'''. Human behavior is organized by how the mind interprets the information it receives.
*'''John Piaget''' - Focused on ''child development''.

===== Behavioral =====
In the behavioural approach, behaviourism originates from the '''learning and observing of other people''' and their responses to different situations. 
*'''B.F. Skinner''' - Used ''pigeons'' and rewarded them when they completed a task successfully. 
*'''Ivan Pavlov''' - Observed behaviour of ''dogs''.  
*'''John B. Watson''' - Performed the ''Little Albert Experiment'', where a baby was scared in order to test learned behaviours)

===== Humanistic =====
In the humanistic approach, a person's behaviour depends on her/his '''goals in life'''. It also makes love a necessary factor in a fruitful life.
*'''Carl Rodgers''' - [[w:Unconditional positive regard|Unconditional positive regard]] [Rodger's Regard].
*'''Abraham Maslov''' - Hierarchy of Needs.

===== Neuroscience/Biological =====
Behaviour originates from '''genetics'''. An example of this is that depression occurs in a human being because of the lack of serotonin.

===== Socioculture =====
Behavior originates from '''societal influences'''. An example of this is if a person is rude to his/her parents, this is because society has considered this practice a norm.

===== Bio-psycho-social =====
This psychological approach consists of three distinct fields. Biological (genetics), psychological (learned; cognitive responses) and social (society influence and expectations). Usually, if a question presents all three fields in a scenario, it is probably biopsychosocial.
====Gestalt====
In the '''Gestalt''' principle, the brain arranges things as a pattern or as a &quot;whole&quot;. This is so that the mind can easily interpret information. These are the principles of Gestalt:
*'''Proximity''' - We combine figures that are in close proximity to each other.
*'''Similarity''' - We combine figures that are similar in features.
*'''Continuity''' - We combine figures that are continuous together vs discontinuous together.
*'''Connectedness''' - We combine figures that are attached together as one.

{{subpage navbar}}
{{CourseCat}}</text>
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    <revision>
      <id>2692044</id>
      <parentid>2692043</parentid>
      <timestamp>2024-12-15T19:39:00Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <minor />
      <comment>/* Study Notes */</comment>
      <origin>2692044</origin>
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      <text bytes="5390" sha1="2xqaca9cfdcvtsvo3qh32l4uyt8asdg" xml:space="preserve">{{:{{BASEPAGENAME}}/Sidebar}}
Introduction to the AP Psychology course and the basic psychological principles needed for the remaining sections in this course.

== Study Notes ==
'''Psychology''': The scientific study of the mind and behavior&lt;br&gt;
'''Empiricism''': A concept from John Locke, in which knowledge in science should be backed up by experimentation and observation. 

The origin of psychology goes back to Ancient Greek, in which philosophers started questioning themselves and human nature. Psychology officially becomes its own science in 1879. 
;Mind's structure/function
*'''Structuralism''' - A school of psychology that teaches &quot;introspection&quot; (looking within) and looks into the mind interpreting subjects as structures (&quot;elements of the mind&quot;). ''Edward Titchener'' is the founder of this school of thought, in which he wanted to discover the immediate thoughts relating to a certain item (immediate feelings about fear: sweaty, fast heart-beat). It is proven to be unreliable as recollections differ between person to person and people are known to recount incorrectly. 
*'''Functionalism''' - A school of psychology that teaches about the performance/function of a certain subject (fear: awakens you, gets you out of danger, more careful, act faster). ''William James'' was inspired by an evolutionist, Charles Darwin, and sought to discover the ways these subjects evoked and allowed us to act. 
;People
[[File:Wilhelm Wundt.jpg|thumb|right|Wilhelm Wundt, the first recorded psychologist]]
'''Wilhem Wundt''' was the first psychologist to establish a psychology lab (in Germany) in 1879. He performed an experiment where:
# The ball is dropped, the lever is pushed.
# The ball is dropped, but the lever is only pushed if the person is ''consciously aware of the ball dropping''. 

Harvard professor '''William James''' published ''Principles of Psychology'' in 1897. He also wrote ''Varieties of Religious Experience'' in 1902, which served as a 'precursor' to the [[Motivation_and_emotion/Book/2024/Subjective_wellbeing_heritability_and_changeability#Biopsychosocial_model|biopsychosocial model]]. 

'''G. Stanley Hall''' was the first president of the APA, first American to receive a degree in psychology and established the first US psychology lab.

'''Mary Whiton Clarkins''': Denied a psychology degree because she's a woman. First female president of the APA.

'''Margaret F. Washburn''': Student of Edward Titchener. The first female to receive a psychology degree. Second female president of the APA and wrote ''The Animal Mind''. 

'''Dorthea Dix''': Wanted more humane treatment of the mentally ill.
===Psychology Branches===
====7 Views of Human Nature====
=====Psychodynamic=====
Also known as &quot;psychoanalytic&quot;, '''psychodynamic''' is the thought in which '''human behavior stems from unconscious desires and behavior from his/her early years (the past)'''. This includes repressed memories and dream interpretation. This thought process originates from ''Sigmund Freud''. For example, a psychologist with the psychodynamic approach might view an outburst as originating from unconscious anger.

===== Cognitive =====
In the cognitive approach, the main target is the way a person '''thinks'''. Human behavior is organized by how the mind interprets the information it receives.
*'''John Piaget''' - Focused on ''child development''.

===== Behavioral =====
In the behavioural approach, behaviourism originates from the '''learning and observing of other people''' and their responses to different situations. 
*'''B.F. Skinner''' - Used ''pigeons'' and rewarded them when they completed a task successfully. 
*'''Ivan Pavlov''' - Observed behaviour of ''dogs''.  
*'''John B. Watson''' - Performed the ''Little Albert Experiment'', where a baby was scared in order to test learned behaviours)

===== Humanistic =====
In the humanistic approach, a person's behaviour depends on her/his '''goals in life'''. It also makes love a necessary factor in a fruitful life.
*'''Carl Rodgers''' - [[w:Unconditional positive regard|Unconditional positive regard]] [Rodger's Regard].
*'''Abraham Maslov''' - Hierarchy of Needs.

===== Neuroscience/Biological =====
Behaviour originates from '''genetics'''. An example of this is that depression occurs in a human being because of the lack of serotonin.

===== Socioculture =====
Behavior originates from '''societal influences'''. An example of this is if a person is rude to his/her parents, this is because society has considered this practice a norm.

===== Bio-psycho-social =====
This psychological approach consists of three distinct fields. Biological (genetics), psychological (learned; cognitive responses) and social (society influence and expectations). Usually, if a question presents all three fields in a scenario, it is probably biopsychosocial.
====Gestalt====
In the '''Gestalt''' principle, the brain arranges things as a pattern or as a &quot;whole&quot;. This is so that the mind can easily interpret information. These are the principles of Gestalt:
*'''Proximity''' - We combine figures that are in close proximity to each other.
*'''Similarity''' - We combine figures that are similar in features.
*'''Continuity''' - We combine figures that are continuous together vs discontinuous together.
*'''Connectedness''' - We combine figures that are attached together as one.

{{subpage navbar}}
{{CourseCat}}</text>
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  <page>
    <title>Possibilities/Curriculum</title>
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      <timestamp>2024-12-15T23:03:46Z</timestamp>
      <contributor>
        <username>Lbeaumont</username>
        <id>278565</id>
      </contributor>
      <comment>Added evolving money</comment>
      <origin>2692096</origin>
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      <text bytes="1691" sha1="fdetvvpfx0zc6tb0whzswsak8q6u6r7" xml:space="preserve">[[File:Vision of the Future.jpg|thumb|200px|Imagine what ''can'' be!]]
This ''possibilities'' curriculum is being developed as part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]] to help people think more creatively, generate alternatives for solving problems, and choose wisely from an abundant set of alternative solutions and actions. 

These courses are available now and establish the core of the curriculum:
*[[Creating Possibilities]]—Navigating problem space
*[[Unleashing Creativity]]—Welcoming new and useful ideas
*[[Thinking Tools]]—Boosting Imagination
*[[Problem Finding]]—Discovering the Real Problem
*[[Solving Problems]]—Creating solutions
*[[Embracing Ambiguity]]—Keep thinking
*[[Transcending Conflict]]—Resolving contradictory goals
*[[Flourishing]]—Realizing human potential
*[[Playing]] — Enjoyable Activity
*[[Envisioning Our Future]] — Describing your vision of our future
*[[Intentional Evolution]] — Choosing our future
*[[Evolving Governments]] — Unleashing collaboration 
*[[Evolving Money]]—Exchanging goods and services 
*[[Creativity]] - Unleashing creativity 

The following courses are proposed but not yet developed:

*Reframing the Problem—Telling a different story
*Beauty&lt;ref&gt;Course needed. Until a course is developed, perhaps the [[What_Matters/Beauty,_awe|Beauty, awe]] topic of the course on ''What Matters'' can be helpful.&lt;/ref&gt;—Savor beauty.
*Aspirations&lt;ref&gt;Course needed. Until a course is developed, perhaps the materials at: http://emotionalcompetency.com/hope.htm can be helpful.&lt;/ref&gt;—Unleash real hope
==References==
&lt;references/&gt;
[[Category:Applied Wisdom]]
[[Category:Curriculum]]
{{Possibilities}}</text>
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    <title>Social Victorians/People/Bourke</title>
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      <text bytes="52287" sha1="81el1p26f3gznyv1yw29buhtqe5emdu" xml:space="preserve">==Also Known As==
* Family name: Bourke [pronounced ''burk'']&lt;ref name=&quot;:62&quot;&gt;{{Cite journal|date=2024-05-07|title=Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Mayo&amp;oldid=1222668659|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Earl_of_Mayo.&lt;/ref&gt;
* The Hon. Algernon Bourke
* Mrs. Guendoline Bourke
* Lady Florence Bourke
* See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the Hon. Algernon Bourke's father.

== Overview ==
Although the Hon. Algernon Henry Bourke was born in Dublin in 1854 and came from a family whose title is in the Peerage of Ireland,&lt;ref name=&quot;:6&quot;&gt;1911 England Census.&lt;/ref&gt; he seems to have spent much of his adult life generally in England and especially in London.

Mrs. Guendoline Bourke was a noted horsewoman and an excellent shot, exhibited at dog shows successfully and was &quot;an appreciative listener to good music.&quot;&lt;ref&gt;&quot;Vanity Fair.&quot; ''Lady of the House'' 15 June 1899, Thursday: 4 [of 44], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004836/18990615/019/0004.&lt;/ref&gt; She was reported as attending many social events without her husband, usually with a quick description of what she wore.

The Hon. Algernon Bourke and Mr. Algernon, depending on the newspaper article, were the same person. Calling him Mr. Bourke in the newspapers, especially when considered as a businessman or (potential) member of Parliament, does not rule out the son of an earl.

== Acquaintances, Friends and Enemies ==

=== Mr. Algernon Bourke ===

* [[Social Victorians/People/Montrose|Marcus Henry Milner]], &quot;one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys&quot;&lt;ref name=&quot;:8&quot;&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 2a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;
* Caroline, Duchess of Montrose — her &quot;legal advisor&quot; on the day of her marriage to Marcus Henry Milner&lt;ref&gt;&quot;Metropolitan Notes.&quot; ''Nottingham Evening Post'' 31 July 1888, Tuesday: 4 [of 4], Col. 1b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000321/18880731/025/0004.&lt;/ref&gt;

== Organizations ==

=== The Hon. Algernon Bourke ===
* Eton
* Cambridge University, Trinity College, 1873, Michaelmas term&lt;ref name=&quot;:7&quot;&gt;Cambridge University Alumni, 1261–1900. Via Ancestry.&lt;/ref&gt;
* Conservative Party
* 1879: Appointed a Poor Law Inspector in Ireland, Relief of Distress Act
* Special Correspondent of The ''Times'' for the Zulu War, accompanying Lord Chelmsford
* White's gentleman's club, St. James's,&lt;ref&gt;{{Cite journal|date=2024-10-09|title=White's|url=https://en.wikipedia.org/wiki/White's|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/White%27s.&lt;/ref&gt; Manager (1897)&lt;ref&gt;&quot;Side Lights on Drinking.&quot; ''Waterford Standard'' 28 April 1897, Wednesday: 3 [of 4], Col. 7a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001678/18970428/053/0003.&lt;/ref&gt;
* Stock Exchange
* Willis's Rooms&lt;blockquote&gt;... the Hon. Algernon Burke [sic], son of the 6th Earl of Mayo, has turned the place into a smart restaurant where choice dinners are served and eaten while a stringed band discourses music. Willis's Rooms are now the favourite dining place for ladies who have no club of their own, or for gentlemen who are debarred by rules from inviting ladies to one of their own clubs. The same gentleman runs a hotel in Brighton, and has promoted several clubs. He has a special faculty for organising places of the kind, without which such projects end in failure.&lt;ref&gt;&quot;Lenten Dullness.&quot; ''Cheltenham Looker-On'' 23 March 1895, Saturday: 11 [of 24], Col. 2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000226/18950323/004/0011. Print p. 275.&lt;/ref&gt;&lt;/blockquote&gt;
*One of the directors, the Franco-English Tunisian Esparto Fibre Supply Company, Ltd.&lt;ref&gt;''Money Market Review'', 20 Jan 1883 (Vol 46): 124.&lt;/ref&gt;
*One of the directors, the Frozen Lake, Ltd., with Admiral Maxse, Lord [[Social Victorians/People/Beresford|Marcus Beresford]], [[Social Victorians/People/Williams|Hwfa Williams]]&lt;ref&gt;&quot;The Frozen Lake, Limited.&quot; ''St James's Gazette'' 08 June 1894, Friday: 15 [of 16], Col. 4a [of 4]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18940608/085/0015. Print p. 15.&lt;/ref&gt;

=== Mr. Algernon Bourke ===

* Head, Messrs. Bourke and Sandys, &quot;that well-known firm of stockbrokers&quot;&lt;ref name=&quot;:8&quot; /&gt;

== Timeline ==
'''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a &quot;convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier.&quot;&lt;ref&gt;{{Cite journal|date=2024-12-01|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Richard_Bourke,_6th_Earl_of_Mayo.&lt;/ref&gt; The Hon. Algernon's brother Dermot became the 7th Earl at 19 years old.

'''1876 November 24, Friday''', the Hon. Algernon Bourke was one of 6 men (2 students, one of whom was Bourke; 2 doctors; a tutor and another man) from Cambridge who gave evidence as witnesses in an inquest about the death from falling off a horse of a student.&lt;ref&gt;&quot;The Fatal Accident to a Sheffield Student at Cambridge.&quot; ''Sheffield Independent'' 25 November 1876, Saturday: 7 [of 12], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000181/18761125/040/0007. Print title: ''Sheffield and Rotherham Independent'', n. p.&lt;/ref&gt;

'''1884 May 3, Saturday''', the &quot;Rochester Conservatives&quot; announced that they would &quot;bring forward the Hon. Algernon Bourke, brother of Lord Mayo, as their second candidate,&quot;&lt;ref&gt;&quot;Election Intelligence.&quot; ''Yorkshire Gazette'' 03 May 1884, Saturday: 4 [of 12], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000266/18840503/011/0004.&lt;/ref&gt; but because he could not be the first candidate, Bourke declined.&lt;ref&gt;&quot;Rochester.&quot; London ''Daily Chronicle'' 09 May 1884, Friday: 3 [of 8], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18840509/049/0003.&lt;/ref&gt;

'''1884 June 18, Wednesday''', Mr. Algernon Bourke was on a committee to watch a [[Social Victorians/Timeline/1884#18 June 1884, Wednesday|Mr. Bishop's &quot;thought-reading&quot; experiment]], which was based on a challenge by Henry Labourchere made the year before. This &quot;experiment&quot; took place before a fashionable audience.

'''1885 October 3, Saturday''', the Hon. Algernon Bourke was named as the Conservative candidate for Clapham in the Battersea and Clapham borough after the Redistribution Bill determined the electoral districts for South London.&lt;ref&gt;&quot;South London Candidates.&quot; ''South London Press'' 03 October 1885, Saturday: 9 [of 16], Col. 5b [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000213/18851003/096/0009. Print p. 9.&lt;/ref&gt; The Liberal candidate, who won, was Mr. J. F. Moulton.

'''1886 July 27, Tuesday''', Algernon Bourke attended a service honoring a memorial at St. Paul's for his father, who had been assassinated.&lt;ref&gt;&quot;Memorial to the Late Earl of Mayo.&quot; ''Northern Whig'' 28 July 1886, Wednesday: 6 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000434/18860728/143/0006. Print p. 6.&lt;/ref&gt;

'''1886 September 2, Thursday''', Mr. Algernon Bourke was part of a group of mostly aristocratic men taking part in [[Social Victorians/Timeline/1886#8 September 1886, Wednesday|a &quot;trial-rehearsal&quot; as part of Augustus Harris's production]] ''A Run of Luck'', about sports.

'''1886 October 2, Saturday''', the Duke of Beaufort and the Hon. Algernon Bourke arrived in Yougal: &quot;His grace has taken a residence at Lismore for a few weeks, to enjoy some salmon fishing on the Blackwater before the close of the season.&quot;&lt;ref&gt;&quot;Chippenham.&quot; ''Wilts and Gloucestershire Standard'' 02 October 1886, Saturday: 8 [of 8], Col. 6a [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001955/18861002/142/0008. Print p. 8.&lt;/ref&gt;

'''1887 December 15''', Hon. Algernon Bourke and Guendoline Stanley were married at St. Paul's, Knightsbridge, by Bourke's uncle the Hon. and Rev. George Bourke. Only family members attended because of &quot;the recent death of a near relative of the bride.&quot;&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 16 December 1887, Friday: 5 [of 8], Col. 7c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18871216/066/0005.&lt;/ref&gt;

'''1888 July 26''', [[Social Victorians/People/Montrose|Caroline Graham Stirling-Crawford]] (known as Mr. Manton for her horse-breeding and -racing operations) and Marcus Henry Milner married.&lt;ref name=&quot;:12&quot;&gt;&quot;Hon. Caroline Agnes Horsley-Beresford.&quot; {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}&lt;/ref&gt; According to the ''Nottingham Evening Post'' of 31 July 1888,&lt;blockquote&gt;LONDON GOSSIP.

(From the ''World''.)

The marriage of &quot;Mr. Manton&quot; was the surprise as well the sensation of last week. Although some wise people noticed a certain amount of youthful ardour in the attentions paid by Mr. Marcus Henry Milner to Caroline Duchess of Montrose at '''Mrs. Oppenheim's ball''', nobody was prepared for the sudden ''dénouement''; '''and it''' were not for the accidental and unseen presence [[Social Victorians/People/Mildmay|a well-known musical amateur]] who had received permission to practice on the organ, the ceremony performed at half-past nine on Thursday morning at St. Andrew's, Fulham, by the Rev. Mr. Propert, would possibly have remained a secret for some time to come. Although the evergreen Duchess attains this year the limit of age prescribed the Psalmist, the bridegroom was only born in 1864. Mr. &quot;Harry&quot; Milner (familiarly known in the City as &quot;Millions&quot;) was one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys, and Mr. Algernon Bourke, the head of the house (who, of course, takes a fatherly interest in the match) went down to Fulham to give away the Duchess. The ceremony was followed by a ''partie carrée'' luncheon at the Bristol, and the honeymoon began with a visit to the Jockey Club box at Sandown. Mr. Milner and the Duchess of Montrose have now gone to Newmarket. The marriage causes a curious reshuffling of the cards of affinity. Mr. Milner is now the stepfather of the [[Social Victorians/People/Montrose|Duke of Montrose]], his senior by twelve years; he is also the father-in-law of [[Social Victorians/People/Lady Violet Greville|Lord Greville]], Mr. Murray of Polnaise, and [[Social Victorians/People/Breadalbane|Lord Breadalbane]].&lt;ref name=&quot;:8&quot; /&gt;&lt;/blockquote&gt;'''1888 December 1st week''', according to &quot;Society Gossip&quot; from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:&lt;blockquote&gt;I am sorry to hear that Mr. Algernon Bourke, who married Miss Sloane-Stanley a short time ago, has been very dangerously ill. Certain complications followed an attack of malarian fever, and last week his mother, the Dowager Lady Mayo, and his brother, Lord Mayo, were hastily summoned to Brighton. Since then a change for the better has taken place, and he is now out of danger.&lt;ref&gt;&quot;Society Gossip. What the ''World'' Says.&quot; ''Hampshire Advertiser'' 08 December 1888, Saturday: 2 [of 8], Col. 5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18881208/037/0002. Print title: ''The Hampshire Advertiser County Newspaper''; print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;'''1889 – 1899 January 1''', the Hon. Algernon Bourke was &quot;proprietor&quot; of White's Club, St. James's Street.&lt;ref name=&quot;:9&quot;&gt;&quot;The Hon. Algernon Bourke's Affairs.&quot; ''Eastern Morning News'' 19 October 1899, Thursday: 6 [of 8], Col. 7c [of7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001152/18991019/139/0006. Print p. 6.&lt;/ref&gt;

'''1889 June 8, Saturday''', the Hon. Algernon Bourke contributed some art he owned to the collection of the Royal Institute of Painters in Water-Colours' [[Social Victorians/Timeline/1889#8 June 1889, Saturday|exhibition of &quot;the works of the 'English Humourists in Art.'&quot;]]

'''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club.

'''1893 February 11, Tuesday''', Algernon Bourke opened Willis's Restaurant:&lt;blockquote&gt;Mr. Algernon Bourke has in his time done many things, and has generally done them well. His recently published history of White's Club is now a standard work. White's Club itself was a few years ago in its agony when Mr. Bourke stepped in and gave it a renewed lease of life. Under Mr. Bourke's auspices &quot;Willis's Restaurant&quot; opened its doors to the public on Tuesday last in a portion of the premises formerly so well known as Willis's Rooms. This new venture is to rival the Amphitryon in the matter of cuisine and wines; but it is not, like the Amphitryon, a club, but open to the public generally. Besides the restaurant proper, there are several ''cabinets particuliers'', and these are decorated with the very best of taste, and contain some fine portraits of the Georges.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Letter from the Linkman.&quot; ''Truth'' 20 April 1893, Thursday: 25 [of 56], Col. 1a [of 2]. ''British Newspaper Archive'' [https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025# https://www.britishnewspaperarchive.co.uk/viewer/bl/0002961/18930420/075/0025]. Print p. 855.&lt;/ref&gt;&lt;/blockquote&gt;'''1893 November 30, Thursday''', with Sir Walter Gilbey the Hon. Algernon Bourke &quot;assisted&quot; in &quot;forming [a] collection&quot; of engravings by George Morland that was exhibited at Messrs. J. and W. Vokins’s, Great Portland-street.&lt;ref&gt;&quot;The George Morland Exhibition at Vokins's.&quot; ''Sporting Life'' 30 November 1893, Thursday: 4 [of 4], Col. 4c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000893/18931130/058/0004.&lt;/ref&gt;

'''1895 February 23, Saturday''', the Hon. Algernon Bourke attended the [[Social Victorians/Timeline/1895#23 February 1895, Saturday|fashionable wedding of Laurence Currie and Edith Sibyl Mary Finch]].

'''1895 August 24, Saturday''', &quot;Marmaduke&quot; in the Graphic says that Algernon Bourke &quot;opened a cyclists' club in Chelsea.&quot;&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 24 August 1895, Saturday: 11 [of 32], Col. 3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18950824/017/0011. Print p. 223.&lt;/ref&gt;

'''1895 October''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]].

'''1896 June 29, Monday''', Algernon Bourke published a letter to the editor of the ''Daily Telegraph'':&lt;blockquote&gt;To the Editor of “The Daily Telegraph.”

Sir — Permit me to make my bow to the public. I am the manager of the Summer Club, which on two occasions bas been the subject of Ministerial interpellation in Parliament. The Summer Club is a small combination, which conceived the idea of attempting to make life more pleasant in London by organising breakfast, luncheon, and teas in Kensington Gardens for its members. This appears to have given offence in some way to Dr. Tanner, with the result that the catering arrangements of the club are now &quot;by order&quot; thrown open to the public. No one is more pleased than I am at the result of the doctor's intervention, for it shows that the idea the Summer Club had of using the parks for something more than mere right of way bas been favourably received. In order, however, that the great British public may not be disappointed, should they all come to lunch at once, I think it necessary to explain that the kitchen, which by courtesy of the lessee of the kiosk our cook was permitted to use, is only 10ft by 5ft; it has also to serve as a scullery and pantry, and the larder, from which our luxurious viands are drawn, is a four-wheeled cab, which comes up every day with the food and returns after lunch with the scraps. Nevertheless, the Summer Club says to the British public — What we have we will share with you, though it don't amount to very much — I am, Sir, your obedient servant,

ALGERNON BOURKE.

White's Club, June 27&lt;ref&gt;&quot;The Summer Club.&quot; ''Daily Telegraph &amp; Courier'' (London) 29 June 1896, Monday: 8 [of 12], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18960629/072/0008. Print title: ''Daily Telegraph'', p. 8.&lt;/ref&gt;&lt;/blockquote&gt;'''1896 July 4, Saturday''', &quot;Marmaduke&quot; in the ''Graphic'' took Bourke's side on the Summer Club in Kensington Park:&lt;blockquote&gt;Most of us have noticed that if we read in the newspapers the account of some matter which we are personally acquainted with the account will generally contain several errors. I have also noticed that when a question is asked in the House of Commons regarding some matter about which I know all the facts the question and the official answer to it frequently contain serious errors. Last week Mr. Akers-Douglas was asked in the House to explain how it was that Mr. Algernon Bourke obtained permission to open the &quot;Summer Club&quot; in Kensington Gardens, and he was questioned upon other particulars connected with the same matter. Both the questions and the official reply showed considerable ignorance of the facts. There has been from time immemorial a refreshment kiosk in Kensington Gardens. Mr. Bourke obtained from the tenant of this permission to use the kitchen for the benefit of the &quot;Summer Club,&quot; and to supply the members of the latter with refreshments. It was a purely commercial transaction. Mr. Bourke then established some wicker seats, a few tables, a tent, and a small hut upon a lawn in the neighbourhood of the kiosk. To do this he must have obtained the permission of Mr. Akers-Douglas, as obviously he would otherwise have been immediately ordered to remove them. Mr. Akers-Douglas equally obviously would not have given his sanction unless he had been previously informed of the objects which Mr. Bourke had in view — to wit, that the latter intended to establish a club there. That being the case, it is difficult to understand for what reason Mr. Akers-Douglas has now decided that any member of the public can use the chairs, tables, and tent belonging to the &quot;Summer Club,&quot; can insist upon the club servants attending upon him, and can compel them to supply him with refreshments. Mr. Akers-Douglas should have thought of the consequences before he granted the permission.&lt;ref&gt;&quot;Marmaduke.&quot; &quot;Court and Club.&quot; The ''Graphic'' 04 July 1896, Saturday: 14 [of 32], Col. 1b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/9000057/18960704/029/0014. Print p. 14.&lt;blockquote&gt;&lt;/blockquote&gt;&lt;/ref&gt;&lt;/blockquote&gt;'''1896 August 10, Monday''', the Morning Leader reported that the Hon. Algernon Bourke, for the Foreign Office, received Li Hung Chang at St. Paul's:&lt;blockquote&gt;At St. Paul's Li Hung was received by Field-Marshal Simmons, Colonel Lane, the Hon. Algernon Bourke, of the Foreign Office (who made the necessary arrangements for the visit) and Canon Newbolt, on behalf of the Dean and Chapter. A crowd greeted Li with a cheer as he drove up in Lord Lonsdale’s striking equipage, and his Excellency was carried up the steps in an invalid chair by two stalwart constables. He walked through the centre door with his suite, and was immediately conducted by Canon Newbolt to General Gordon’s tomb in the north aisle, where a detachment of boys from the Gordon Home received him as a guard of honor.

Li inspected the monument with marked interest, and drew the attention of his suite to the remarkable likeness to the dead hero. He laid a handsome wreath of royal purple asters, lilies, maidenhair fern, and laurel, tied with a broad band of purple silk, on the tomb.

The visit was not one of inspection of the building, but on passing the middle aisle the interpreter called the attention of His Excellency to the exquisite architecture and decoration of the chancel. Li shook hands in hearty English fashion with Canon Newbolt and the other gentlemen who had received him, and, assisted by his two sons, walked down the steps to his carriage. He returned with his suite to Carlton House-terrace by way of St. Paul’s Churchyard, Cannon-st., Queen Victoria-st., and the Embankment.&lt;ref&gt;&quot;At St. Paul's.&quot; ''Morning Leader'' 10 August 1896, Monday: 7 [of 12], Col. 2b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18960810/134/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;'''1896 August 19, Wednesday''', the ''Edinburgh Evening News'' reported on the catering that White's Club and Mr Algernon Bourke arranged for the visiting Li Hung Chang:&lt;blockquote&gt;It is probably not generally known (says the &quot;Chef&quot;) that Mr Algernon Bourke, manager of White's Club, London, has undertaken to the whole of the catering for our illustrious visitor front the Flowery Land. Li Hung Chang has five native cooks in his retinue, and the greatest good fellowship exists between them and their English ''confreres'', although considerable difficulty is experienced in conversation in understanding one another's meaning. There are between 40 and and 50 to cater for daily, besides a staff about 30; that Mr Lemaire finds his time fully occupied. The dishes for his Excellency are varied and miscellaneous, and from 14 to 20 courses are served at each meal. The bills of fare contain such items as bird's-nest soup, pigs' kidneys stewed in cream, boiled ducks and green ginger, sharks' fins, shrinips and prawns stewed with leeks and muscatel grapes, fat pork saute with peas and kidney beans. The meal usually winds with fruit and sponge cake, and freshly-picked green tea as liqueur.&lt;ref&gt;&quot;Li Hung Chang's Diet.&quot; ''Edinburgh Evening News'' 19 August 1896, Wednesday: 3 [of 4], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18960819/057/0003.&lt;/ref&gt;&lt;/blockquote&gt;

'''1896 November 6, Friday''', Algernon Bourke was on the committee for [[Social Victorians/Timeline/1896#1896 November 6, Friday|the Prince's Club ice-skating rink, which opened on this day]].&lt;p&gt;

'''1896 November 25, Wednesday''', Mr. and Mrs. Algernon Bouke attended [[Social Victorians/Timeline/1896#23 November 1896, Monday23 November 1896, Monday|Lord and Lady Burton's party for Derby Day]].&lt;p&gt;
'''1896 December 4, Friday''', the Orleans Club at Brighton was robbed:&lt;blockquote&gt;The old building of the Orleans Club at Brighton, which opens its new club house at 33, Brunswick-terrace to-day, was the scene of a very ingenious burglary during the small hours of yesterday morning. The greater portion of the club property had already been removed to the new premises, but Mr Algernon Bourke, his private secretary, and some of the officials of the club, still occupied bed-rooms at the house in the King’s-road. The corner shop of the street front is occupied by Mr. Marx, a jeweller in a large way of business, and upon his manager arriving at nine o'clock he discovered that the place had been entered through hole in the ceiling, and a great part of a very valuable stock of jewelry extracted. An examination of the morning rooms of the club, which runs over Mr. Marx's establishment reveal a singularly neat  specimen of the burglar's art. A piece of the flooring about 15in square had been removed by a series of holes bored side by side with a centre-bit, at a spot where access to the lofty shop was rendered easy by a tall showcase which stood convemently near. A massive iron girder had been avoided by a quarter of an inch, and this circumstance and the general finish of the operation point to an artist in his profession, who had acquired an intimate knowledge of the premises. The club doors were all found locked yesterday morning, and the means of egress adopted by the thief are at present a mystery.&lt;ref&gt;&quot;Burglary at Brighton.&quot; ''Daily Telegraph &amp; Courier'' (London) 05 December 1896, Saturday: 5 [of 12], Col. 7a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/18961205/090/0005. Print title: ''Daily Telegraph''; p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

'''1897 July 2, Friday''', the Hon. A. and Mrs. A. Bourke and Mr. and Mrs. Bourke attended the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]] at Devonshire House.&lt;p&gt;

'''23 July 1897 — or 30 July 1897 – Friday''', Guendonline Bourke attended [[Social Victorians/Timeline/1897#23 July 1897, Friday|Lady Burton's party at Chesterfield House]]. &lt;blockquote&gt;Far the prettiest women in the room were Lady Henry Bentinck (who looked perfectly lovely in pale yellow, with a Iong blue sash; and Mrs. Algernon Bourke, who was as smart as possible in pink, with pink and white ruchings on her sleeves and a tall pink feather in her hair.&lt;ref&gt;&quot;Lady Burton's Party at Chesterfield House.&quot; ''Belper &amp; Alfreton Chronicle'' 30 July 1897, Friday: 7 [of 8], Col. 1c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004151/18970730/162/0007. Print title: ''Belper and Alfreton Chronicle''; n.p.&lt;/ref&gt;&lt;/blockquote&gt;

'''1898 January 5, Wednesday''', the ''Irish Independent'' reported that &quot;Mr Algernon Bourke, the aristocratic stock broker ... was mainly responsible for the living pictures at the Blenheim Palace entertainment.&lt;ref&gt;&quot;Mr Algernon Bourke ....&quot; ''Irish Independent'' 05 January 1898, Wednesday: 6 [of 8], Col. 2c [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001985/18980105/115/0006.&lt;/ref&gt;&lt;p&gt;

'''1899 January 10, Tuesday''', the Brighton Championship Dog Show opened:&lt;blockquote&gt;Princess of Wales a Winner at the Ladies’ Kennel Club Show.

[Exclusive to &quot;The Leader.&quot;)

The Brighton Championship Dog Show opened in the Dome and Corn Exchange yesterday, and was very well patronised by visitors and exhibitors. Among the latter was H.R.H. the Princess of Wales, who did very well; and others included Princess Sophie Duleep Singh, Countess De Grey, Sir Edgar Boehm, the Hon Mrs. Algernon Bourke, Lady Cathcart, Lady Reid, Mr. Shirley (chairman of the Kennel Club), and the Rev. Hans Hamiiton (president of the Kennel Club).

The entry of bloodhounds is one of the best seen for some time; the Great Danes are another stronyg lot; deerhounds are a fine entry, all good dogs, and most of the best kennels represented; borzois are another very stylish lot. The bigger dogs are, as usual, in the Corn Exchange and the &quot;toy&quot; dogs in the Dome. To everyone's satsfaction the Princess of Wales carried off two first prizes with Alex in the borzois class.&lt;ref&gt;&quot;Dogs at Brighton.&quot; ''Morning Leader'' 11 January 1899, Wednesday: 8 [of 12], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004833/18990111/142/0008. Print p. 8.&lt;/ref&gt;&lt;/blockquote&gt;'''1899 June 1, Thursday''', the Hon. Algernon and Guendoline Bourke attended the wedding of her brother, Sloane Stanley and Countess Cairns at Holy Trinity Church, Brompton.&lt;ref&gt;&quot;Marriage of Mr. Sloane Stanley and Countess Cairns.&quot; ''Hampshire Advertiser'' 03 June 1899, Saturday: 6 [of 8], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000495/18990603/049/0006. Print p. 6.&lt;/ref&gt;&lt;p&gt;

'''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:&lt;blockquote&gt;The public examination of the Hon. Algernon Bourke was held before Mr Registrar Giffard yesterday, at the London Bankruptcy Court. The debtor, described as proprietor of a St. James's-street club, furnished a statement of affairs showing unsecured debts £13,694 and debts fully secured £12,800, with assets which are estimated at £4,489 [?]. He stated, in reply to the Official Receiver, that he was formerly a member of the Stock Exchange, but had nothing to do with the firm of which he was a member during the last ten years. He severed his connection with the firm in May last, and believed he was indebted to them to the extent of £2,000 or £3,000. He repudiated a claim which they now made for £37,300. In 1889 he became proprietor of White's Club, St. James's-street, and carried it on until January 1st last, when he transferred it to a company called Recreations, Limited. One of the objects of the company was to raise money on debentures. The examination was formally adjourned.&lt;ref name=&quot;:9&quot; /&gt;&lt;/blockquote&gt;'''1899 November 8, Wednesday''', the Hon. Algernon Bourke's bankruptcy case came up again:&lt;blockquote&gt;At Bankruptcy Court, yesterday, the case the Hon. Algernon Bourke again came on for hearing before Mr. Registrar Giffard, and the examination was concluded. The debtor has at various times been proprietor of White’s Club, St. James’s-street, and the Orleans’ Club, Brighton, and also of Willis's Restaurant, King-street, St. James's. He attributed his failure to losses sustained by the conversion of White’s Club and the Orleans' Club into limited companies, to the payment of excessive Interest on borrowed money, and other causes. The liabilities amount to £26,590, of which £13,694 are stated to be unsecured, and assets £4,409.&lt;ref&gt;&quot;Affairs of the Hon. A. Bourke.&quot; ''Globe'' 09 November 1899, Thursday: 2 [of 8], Col. 1c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18991109/020/0002. Print p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

'''1899 December 23, Saturday''', &quot;Mr. Algernon Bourke has departed for a tour in Africa, being at present the guest of his brother in Tunis.&quot;&lt;ref&gt;&quot;The Society Pages.&quot; ''Walsall Advertiser'' 23 December 1899, Saturday: 7 [of 8], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001028/18991223/143/0007. Print p. 7.&lt;/ref&gt;

'''1900 February 15, Thursday''', Miss Daphne Bourke, the four-year-old daughter of the Hon. Algernon and Mrs. Bourke was a bridesmaid in the wedding of Enid Wilson and the Earl of Chesterfield, so presumably her parents were present as well.&lt;ref&gt;&quot;London Day by Day.&quot; ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph &amp; Courier'' (London). Print p. 8.&lt;/ref&gt;

'''1900 September 16''', the Hon. Algernon Bourke became the heir presumptive to the Earldom of Mayo when his older brother Captain Hon. Sir Maurice Archibald Bourke died.

'''1900 October 06, Saturday''', the ''Weekly Irish Times'' says that Mr. Algernon Bourke, now heir presumptive to the earldom of Mayo, &quot;has been for some months lately staying with Mr. Terence Bourke in Morocco.&quot;&lt;ref&gt;&quot;Society Gossip.&quot; ''Weekly Irish Times'' 06 October 1900, Saturday: 14 [of 20], Col. 3b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001684/19001006/121/0014. Print p. 14.&lt;/ref&gt;

'''1901 May 30, Thursday''', the Hon. Mrs. Algernon Bourke attended the fashionable Ladies' Kennel Association Dog Show.

'''1901 July 4, Thursday''', Guendoline and Daphne Bourke attended a children's party hosted by the Countess of Yarborough:&lt;blockquote&gt;The Countess of Yarborough gave a charming children's party on Thursday (4th) afternoon at her beautiful house in Arlington Street. The spacious ballroom was quite filled with little guests and their mothers. Each little guest received a lovely present from their kind hostess. The Duchess of Beaufort, in grey, and with a large black picture hat, brought her two lovely baby girls, Lady Blanche and Lady Diana Somerset, both in filmy cream [Col. 2b–3a] lace frocks. Lady Gertrude Corbett came with her children, and Ellen Lady Inchiquin with hers. Lady Southampton, in black, with lovely gold embroideries on her bodice, brought her children, as also did Lady Heneage and Mr. and Lady Beatrice Kaye. Lady Blanche Conyngham, in écru lace, over silk, and small straw hat, was there; also Mrs. Smith Barry, in a lovely gown of black and white lace. The Countess of Kilmorey, in a smart grey and white muslin, brought little Lady Cynthia Needham, in white; Mrs. Arthur James, in black and white muslin; and the Countess of Powys, in mauve silk with much white lace; Lady Sassoon, in black and white foulard; Victoria Countess of Yarborough, came on from hearing Mdme. Réjane at Mrs. Wernher's party at Bath House; and there were also present Lord Henry Vane-Tempest, the Earl of Yarborough, Lady Naylor-Leyland's little boys; the pretty children of Lady Constance Combe, Lady Florence Astley and her children, and Lady Meysey Thompson (very smart in mauve and white muslin) with her children; also Hon. Mrs. Algernon Bourke, in pale grey, with her pretty little girl.&lt;ref&gt;&quot;The Countess of Yarborough ....&quot; ''Gentlewoman'' 13 July 1901, Saturday: 76 [of 84], Col. 2b, 3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010713/381/0076. Print p. xxxvi.&lt;/ref&gt;&lt;/blockquote&gt;'''20 July 1901, Saturday''', the ''Gentlewoman'' published the Hon. Mrs. Algernon Bourke's portrait (identified with &quot;Perthshire&quot;) in its 3rd series of &quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot;&lt;ref&gt;&quot;The Great County Sale at Earl's Court. Portraits of Stallholders.&quot; ''Gentlewoman'' 20 July 1901, Saturday: 31 [of 60], Col. 4b [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/19010720/141/0031. Print n.p.&lt;/ref&gt; Their daughter Daphne appears in the portrait as well.

'''1901 September 12, Thursday''', Mrs. Guendoline Bourke's name is listed as Gwendolen Bourke, but the spelling is not what she objected to:&lt;blockquote&gt;Mr. Underhill, the Conservative agent, mentioned to the Revising Barrister (Mr. William F. Webster) that the name of the Hon. Mrs. Gwendolen Bourke was on the list in respect of the house, 75, Gloucester-place. The lady had written to him to say that she was the Hon. Mrs. Algernon Bourke and that she wished that name to appear on the register. In reply to the Revising Barrister, Mr. Underhill said that “Algernon” was the name the lady’s husband.

Mr. Cooke, the rate-collector, said that Mrs. Bourke had asked to be addressed Mrs. Algernon Bourke, but that the Town Clerk thought the address was not a correct one. The lady signed her cheques Gwendolen.”

Mr. Underhill said the agents frequently had indignant letters from ladies because they were not addressed by their husband’s Christian name.

The Revising Barrister — lf a lady gave me the name of Mrs. John Smith I should say I had not got the voter’s name. The name Gwendolen must remain.&lt;ref&gt;&quot;Ladies’ Names.&quot; ''Morning Post'' 12 September 1901, Thursday: 7 [of 10], Col. 3a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19010912/130/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;'''1902 September 4, Thursday''', the Daily Express reported that &quot;Mrs. Algernon Bourke is staying with Lord and Lady Alington at Scarborough.&quot;&lt;ref&gt;&quot;Onlooker.&quot; &quot;My Social Diary.&quot; &quot;Where People Are.&quot; ''Daily Express'' 04 September 1902, Thursday: 5 [of 8], Col. 1b? [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020904/099/0005. Print p. 4, Col. 7b [of 7].&lt;/ref&gt;

'''1902 October 24, Friday''', the Hon. Algernon Bourke [[Social Victorians/Timeline/1900s#24 October 1902, Friday|opened the Prince's ice-skating rink for the season]], which he had been doing since 1895.

'''1902 December 9, Tuesday''', Guendonline Bourke attended [[Social Victorians/Timeline/1900s#9 December 1902, Tuesday|Lady Eva Wyndham-Quin's &quot;at home,&quot; held at the Welch Industrial depot]] for the sale Welsh-made Christmas gifts and cards. Bourke wore &quot;a fur coat and a black picture hat.&quot;&lt;ref&gt;&quot;A Lady Correspondent.&quot; &quot;Society in London.&quot; ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.&lt;/ref&gt;

'''1905 February 17, Friday''', the Dundee ''Evening Post'' reported that Algernon Bourke &quot;set up a shop in Venice for the sale of art treasures and old furniture.&quot;&lt;ref&gt;&quot;Social News.&quot; Dundee ''Evening Post'' 17 February 1905, Friday: 6 [of 6], Col. 7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000582/19050217/105/0006. Print p. 6.&lt;/ref&gt;

'''1905, last week of July''', Guendoline Bourke and daughter Daphne Bourke — who was 10 years old — attended [[Social Victorians/Timeline/1900s#Last week of July, 1905|Lady Cadogan's children's party at Chelsea House]]. Daphne was &quot;One of loveliest little girls present.&quot;&lt;ref&gt;&quot;Court and Social News.&quot; ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.&lt;/ref&gt;

'''1913 May 7, Wednesday''', Guendoline Bourke presented her daughter Daphne Bourke at court:&lt;blockquote&gt;Mrs. Algernon Bourke presented her daughter, and wore blue and gold broché with a gold lace train.&lt;ref&gt;&quot;Social and Personal.&quot; London ''Daily Chronicle'' 08 May 1913, Thursday: 6 [of 12], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/19130508/120/0006. Print p. 6.&lt;/ref&gt;&lt;/blockquote&gt;The ''Pall Mall Gazette'' has a description of Daphne Bourke's dress:&lt;blockquote&gt;Court dressmakers appear to have surpassed all previous records in their efforts to make the dresses for to-night’s Court as beautiful as possible. Noticeable among these is the dainty presentation gown to be worn by Miss Bourke, who will be presented by her mother, the Hon. Mrs. Algernon Bourke. This has a skirt of soft white satin draped with chiffon paniers and a bodice veiled with chiffon and trimmed with diamanté and crystal embroidery. Miss Bourke’s train, gracefully hung from the shoulders, is of white satin lined with pale rose pink chiffon and embroidered with crystal and diamanté.&lt;ref&gt;&quot;Fashion Day by Day. Lovely Gowns for To-night's Court.&quot; ''Pall Mall Gazette'' 07 May 1913, Wednesday: 13 [of 18], Col. 1a [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/19130507/199/0013. Print n.p.&lt;/ref&gt;&lt;/blockquote&gt;

== Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball ==
According to both the ''Morning Post'' and the ''Times'', the Hon. Algernon Bourke was among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].&lt;ref name=&quot;:2&quot; /&gt;&lt;ref name=&quot;:3&quot; /&gt; Based on the people they were dressed as, Guendonine Bourke was probably in this procession but it seems unlikely that Algernone Bourke was.

[[File:Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in an historical costume with a headdress and a very large fan|Hon. Guendoline Bourke as Salammbô. ©National Portrait Gallery, London.]]
=== Hon. Guendoline Bourke ===
[[File:Alfons Mucha - 1896 - Salammbô.jpg|thumb|left|alt=Highly stylized orange-and-yellow painting of a bare-chested woman with a man playing a harp at her feet|Alfons Mucha's 1896 ''Salammbô''.]]
Lafayette's portrait (right) of &quot;Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô&quot; in costume is photogravure #128 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot;&gt;&quot;Devonshire House Fancy Dress Ball (1897): photogravures by Walker &amp; Boutall after various photographers.&quot; 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.&lt;/ref&gt; The printing on the portrait says, &quot;The Hon. Mrs. Algernon Bourke as Salammbo.&quot;&lt;ref&gt;&quot;Mrs. Algernon Bourke as Salammbo.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158491/Guendoline-Irene-Emily-Bourke-ne-Sloane-Stanley-as-Salammb.&lt;/ref&gt;

==== Newspaper Accounts ====
The Hon. Mrs. A. Bourke was dressed as

* Salambo in the Oriental procession.&lt;ref name=&quot;:2&quot;&gt;&quot;Fancy Dress Ball at Devonshire House.&quot; ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.&lt;/ref&gt;&lt;ref name=&quot;:3&quot;&gt;&quot;Ball at Devonshire House.&quot; The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.&lt;/ref&gt;
* &quot;(Egyptian Princess), drapery gown of white and silver gauze, covered with embroidery of lotus flowers; the top of gown appliqué with old green satin embroidered blue turquoise and gold, studded rubies; train of old green broché.&quot;&lt;ref&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 40, Col. 3a}}
*&quot;Mrs. A. Bourke, as an Egyptian Princess, with the Salambo coiffure, wore a flowing gown of white and silver gauze covered with embroidery of lotus flowers. The top of the gown was ornamented with old green satin embroidered with blue turquoise and gold, and studded with rubies. The train was of old green broché with sides of orange and gold embroidery, and from the ceinture depended long bullion fringe and an embroidered ibis.&quot;&lt;ref&gt;“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.&lt;/ref&gt;{{rp|p. 3, Col. 3b}}

==== Salammbô ====
Salammbô is the eponymous protagonist in Gustave Flaubert's 1862 novel.&lt;ref name=&quot;:5&quot;&gt;{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&amp;oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.&lt;/ref&gt; Ernest Reyer's opera ''Salammbô'' was based on Flaubert's novel and published in Paris in 1890 and performed in 1892&lt;ref&gt;{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&amp;oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.&lt;/ref&gt; (both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well&lt;ref name=&quot;:5&quot; /&gt;). Alfons Mucha's 1896 lithograph of Salammbô was published in 1896, the year before the ball (above left).[[File:Algernon Henry Bourke Vanity Fair 20 January 1898.jpg|thumb|alt=Old colored drawing of an elegant elderly man dressed in a 19th-century tuxedo with a cloak, top hat and formal pointed shoes with bows, standing facing 1/4 to his right|''Algy'' — Algernon Henry Bourke — by &quot;Spy,&quot; ''Vanity Fair'' 20 January 1898]]
=== Hon. Algernon Bourke ===
[[File:Hon-Algernon-Henry-Bourke-as-Izaak-Walton.jpg|thumb|left|alt=Black-and-white photograph of a man richly dressed in an historical costume sitting in a fireplace that does not have a fire and holding a tankard|Hon. Algernon Henry Bourke as Izaak Walton. ©National Portrait Gallery, London.]]
'''Lafayette's portrait''' (left) of &quot;Hon. Algernon Henry Bourke as Izaak Walton&quot; in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.&lt;ref name=&quot;:4&quot; /&gt; The printing on the portrait says, &quot;The Hon. Algernon Bourke as Izaak Walton.&quot;&lt;ref&gt;&quot;Hon. Algernon Bourke as Izaak Walton.&quot; ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.&lt;/ref&gt;

This portrait is amazing and unusual: Algernon Bourke is not using a photographer's set with theatrical flats and props, certainly not one used by anyone else at the ball itself. Isaak Walton (baptised 21 September 1593 – 15 December 1683) wrote ''The Compleat Angler''.&lt;ref&gt;{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&amp;oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.&lt;/ref&gt; A cottage Walton lived in and willed to the people of Stafford was photographed in 1888, suggesting that its relationship to Walton was known in 1897, raising a question about whether Bourke could have used the fireplace in the cottage for his portrait. (This same cottage still exists, as the [https://www.staffordbc.gov.uk/izaak-waltons-cottage Isaak Walton Cottage] museum.)

A caricature portrait (right) of the Hon. Algernon Bourke, called &quot;Algy,&quot; by Leslie Ward (&quot;Spy&quot;) was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its &quot;Men of the Day&quot; series,&lt;ref&gt;{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&amp;oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).&lt;/ref&gt; giving an indication of what he looked like out of costume.

=== Mr. and Mrs. Bourke ===
The ''Times'' made a distinction between the Hon. Mr. and Mrs. A. Bourke and Mr. and Mrs. Bourke, including both in the article.&lt;ref name=&quot;:3&quot; /&gt; Occasionally this same article mentions the same people more than once in different contexts and parts of the article, so they may be the same couple. (See [[Social Victorians/People/Bourke#Notes and Question|Notes and Question]] #2, below.)

== Demographics ==
*Nationality: Anglo-Irish&lt;ref&gt;{{Cite journal|date=2020-11-14|title=Richard Bourke, 6th Earl of Mayo|url=https://en.wikipedia.org/w/index.php?title=Richard_Bourke,_6th_Earl_of_Mayo&amp;oldid=988654078|journal=Wikipedia|language=en}}&lt;/ref&gt;
*Occupation: journalist. 1895: restaurant, hotel and club owner and manager&lt;ref&gt;''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.&lt;/ref&gt;
=== Residences ===
*Ireland: 1873: Palmerston House, Straffan, Co. Kildare.&lt;ref name=&quot;:7&quot; /&gt; Not Co. Mayo?
*1890: 33 Cadogan Terrace
*1891: 33 Cadogan Terrace, Kensington and Chelsea, a dwelling house&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.&lt;/ref&gt;
*1894: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.&lt;/ref&gt;
*1900: 181 Pavilion Road, Kensington and Chelsea&lt;ref&gt;Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.&lt;/ref&gt;
*1911: 1911 Fulham, London&lt;ref name=&quot;:6&quot; /&gt;
*20 Eaton Square, S.W. (in 1897)&lt;ref name=&quot;:0&quot;&gt;{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. &amp; C. Black|language=en}} 712, Col. 1b.&lt;/ref&gt; (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]])

== Family ==
*Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)&lt;ref&gt;&quot;Hon. Algernon Henry Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
*Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)&lt;ref name=&quot;:1&quot;&gt;&quot;Guendoline Irene Emily Stanley.&quot; {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}&lt;/ref&gt;
#Daphne Marjory Bourke (5 April 1895 – 22 May 1962)

=== Relations ===
*Hon. Algernon Henry Bourke (the 3rd son of the [[Social Victorians/People/Mayo|6th Earl of Mayo]]) was the older brother of Lady Florence Bourke.&lt;ref name=&quot;:0&quot; /&gt;

==== Other Bourkes ====
*Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)&lt;ref&gt;&quot;Hubert Edward Madden Bourke-Borrowes.&quot; {{Cite web|url=https://www.thepeerage.com/p52401.htm#i524004|title=Person Page|website=www.thepeerage.com|access-date=2021-08-25}} https://www.thepeerage.com/p52401.htm#i524004.&lt;/ref&gt;
*Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;&lt;ref&gt;&quot;Lady Eva Constance Aline Bourke.&quot; {{Cite web|url=https://www.thepeerage.com/p2575.htm#i25747|title=Person Page|website=www.thepeerage.com|access-date=2020-12-02}} https://www.thepeerage.com/p2575.htm#i25747.&lt;/ref&gt; he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926.
== Writings, Memoirs, Biographies, Papers ==

=== Writings ===
* Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892.
* Bourke, the Hon. Algernon, ed., &quot;with a brief Memoir.&quot; ''Correspondence of Mr Joseph Jekyll with His Sister-in-Law, Lady Gertrude Sloane Stanley, 1818–1838''. John Murray, 1893.
* Bourke, the Hon. Algernon, ed. ''Correspondence of Mr Joseph Jekyll''. John Murray, 1894.

=== Papers ===
* Where are the papers for the Earl of Mayo family? Are Algernon Bourke's papers with them?

== Notes and Questions ==
#The portrait of Algernon Bourke in costume as Isaac Walton is really an amazing portrait with a very interesting setting, far more specific than any of the other Lafayette portraits of these people in their costumes. Where was it shot? Lafayette is given credit, but it's not one of his usual backdrops. If this portrait was taken the night of the ball, then this fireplace was in Devonshire House; if not, then whose fireplace is it?
#The ''Times'' lists Hon. A. Bourke (at 325) and Hon. Mrs. A. Bourke (at 236) as members of a the &quot;Oriental&quot; procession, Mr. and Mrs. A. Bourke (in the general list of attendees), and then a small distance down Mr. and Mrs. Bourke (now at 511 and 512, respectively). This last couple with no honorifics is also mentioned in the report in the London ''Evening Standard'', which means the Hon. Mrs. A. Bourke, so the ''Times'' may have repeated the Bourkes, who otherwise are not obviously anyone recognizable. If they are not the Hon. Mr. and Mrs. A. Bourke, then they are unidentified. It seems likely that they are the same, however, as the newspapers were not perfectly consistent in naming people with their honorifics, even in a single story, especially a very long and detailed one in which people could be named more than once.
#Three slightly difficult-to-identify men were among the Suite of Men in the [[Social Victorians/1897 Fancy Dress Ball/Quadrilles Courts#&quot;Oriental&quot; Procession|&quot;Oriental&quot; procession]]: [[Social Victorians/People/Halifax|Gordon Wood]], [[Social Victorians/People/Portman|Arthur B. Portman]] and [[Social Victorians/People/Sarah Spencer-Churchill Wilson|Wilfred Wilson]]. The identification of Gordon Wood and Wilfred Wilson is high because of contemporary newspaper accounts. The Hon. Algernon Bourke, who was also in the Suite of Men, is not difficult to identify at all. Arthur Portman appears in a number of similar newspaper accounts, but none of them mentions his family of origin.
#[http://thepeerage.com The Peerage] has no other Algernon Bourkes.
#The Hon Algernon Bourke is #235 on the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of people who were present]]; the Hon. Guendoline Bourke is #236; a Mr. Bourke is #703; a Mrs. Bourke is #704.
== Footnotes ==
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    <title>Social Victorians/Timeline/1900s</title>
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      <text bytes="105906" sha1="ox1bajze6hxobepeu1woekuds07dio7" xml:space="preserve">[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s]] [[Social Victorians/Timeline/1890s | 1890s]] 1900s

==1900==
1900, early, [[Social Victorians/People/Mathers|MacGregor and Moina Mathers]] were living at 87 Rue Mozart, Paris (Howe 203).
===January 1900===
====1 January 1900, Monday, New Year's Day====
====13 January 1900, Tuesday====
&lt;blockquote&gt;THE HOUSEHOLD TROOPS. ENTERTAINMENT AT HER MAJESTY'S.

The Prince and Princess of Wales, accompanied by Princess Victoria and Prince Charles of Denmark, attended the entertainment to aid the widows and orphans of her Majesty's Household Troops, organised by Mrs. Arthur Paget and presented under the direction of Mr. H. Beerbohm Tree at Her Majesty's Theatre last night.

... [The major part of this story is the program of the entertainment, in which [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], among others, played an important part.]

Among those present at the entertainment were: The Prince and Princess of Wales, Princess Victoria of Wales, and Prince Charles of Denmark, the French Ambassador, the Russian Ambassador, the Portuguese Minister, Count Mensdorff, the Austrian Embassy, Prince and Princess Demidoff, Prince and Princess Hatzfeldt, Prince and Princess Alexis Dolgorouki, Count and Countess Roman Potocki, Count and Countess Alexander Münister, the Duke and Duchess of Devonshire, the Marquis of Downshire, the Earl and Countess of Cork, the Earl and Countess of Westmorland, the Earl and Countess of Gosford, the Earl of Lathom, the Countess of Ancaster, the Countess of Wilton, the Countess of Yarborough, the Countess of Huntingdon, Viscount Curzon, Lord and Lady Farquhar, Lord and Lady Savile, Lord Rowton, Lord Westbury, Baroness d'Erlanger, Count and Countess Seilern, Lord and Lady Ribblesdale, Lord and Lady Hothfield, Lord and Lady Raincliffe, Lord Wandsworth, Lord Charles Montagu, Lady Cunard, Sir Edgar and Lady Helen Vincent, Lady Kathleen and Mr. Pilkington, Lady Violet Brassey, Lady Grey Egerton, the Hon. Humphry and Lady Feodorowna Sturt, Lady Ripley, Lady Katherine Coke, Lady Agneta Montagu, Lady Tatton Sykes, Lady Templemore, Lady Florence Grant, Lady Garrick, Lady Pearson, Lady Constance Haddon, Sir F. Burdett, the Hon. M. Charteris, Sir A. de la Rue, Sir Frederick and Lady Milner, the Hon. E. Stonor, Sir Edward and Lady Sassoon, Mrs. Joseph Chamberlain, the Hon. Mrs. Lawrence, the Hon. Mrs. Napier, Sir Charles Forbes, Mrs. Bradley Martin, Mrs. Cornwallis West, Mr. Arnold Morley, Mr. L. Neumann, Madame Vagliano, Mr. Gillett, Mrs. Godfrey Samuelson, Mrs. Reginald Ward, Mr. and Mrs. Arthur Wilson, Mr. Menzies, Mr. Dreyfous [sic], Mrs. George Coats, Mr. Hartmann, Mrs. Rube, Mrs. Neumann, Mr. Lukach, Mrs. Candy, Mr. Bargrave Deane, Mr. L. V. Harcourt, Mrs. Oppenheim, Mrs. Lionel Phillips, Mr. King. Mr. James Finch, Mrs. Clayton Glyn, Miss Van Wart, Mr. Hall Walker, Mr. Drexell, Mrs. Van Raalte, Mr. Alfred Beit, Mr. Douglas Uzielli, Mrs. Alfred Harmsworth, Mr. Munday, Mrs. William James, Mrs. Newhouse, Mrs. Max Waechter, Mr. G. Prentis, Mrs. M'Calmont, Mr. Blacklock, Mrs. Ausell, Captain Holford (Equerry to the Prince of Wales), Mr. De Nino, Mrs. Keyser, Mrs. Fleming, Mrs. Breitmeyer, Mrs. Wernher, Mrs. Armour, Mr. Van Alan, Mrs. Ewart, Mrs. Carl Meyer, Mrs. Powell, Mr. Hambro, Colonel Charles Allen, Colonel Cunningham, Mrs.Hutchinson, Mrs. Schumacher, Colonel Kennard, Mrs. Fludyer, Mrs. Williamson, Mr. Thellusson, Mr. Sackville West, Captain M'Neil, Mrs. Dalrymple Hamilton, Mrs. Penn Curzon, Mrs. Hamar Bass, Mrs. Kuhliug, General Stracey, Mrs. Jeffcock, Colonel Thynne. (1900-02-14 Morning Post).&lt;/blockquote&gt;

====17 January 1900, Saturday====

1900 February 17, Lady Greville writes about the amateur theatricals Muriel Wilson is involved in: &lt;blockquote&gt;The most notable social event of the week was the amateur performance of tableaux at Her Majesty's Theatre. One is accustomed to the amateurs under every aspect, leaping in where angels fear to tread, essaying the most difficult parts, dabbling in the arts of music and literature, but so full and rich and interesting a performance has rarely been given before. To begin with, there was a masque, modelled on the Elizabethan lines, with song and dance, and special music composed for the occasion by Mr. Hamish McCunn, dresses statuesque and graceful, and a bevy of pretty women to carry out the idea.

One original feature there was, too, which certainly did not present itself before our Virgin Queen, and that was the graceful fencing of Miss Lowther, who looked an ideal young champion in her russet suit and jaunty little cap. A very young debutante appeared in the person of Miss Viola Tree, who, dressed in the nest diaphanous garments, acted with a grace and lightness that promises well for her future career. Mrs. Crutchly, as &quot;Glory,&quot; appeared amid a din of thunder and a rosy glare of limelight, and clashed her cymbals in truly determined fashion. An element of wildness suited to the character, distinguished her agreeable posturing, and her high spiked crown gave distinct individuality to the representation. Mrs. Martineau, Hebe-like in a white robe and a large crown of roses, as if she had just stepped out of a picture by Leighton, then danced and took the palm for poetry and suppleness of movement; Miss [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], meanwhile, having daringly shot up through a trap-door in scarlet robes with a flaming torch, announced herself as &quot;War,&quot; and beckoned to Glory, Victory, and Prosperity, when they finished their performance, to sit beside her on her throne. &quot;Rumour,&quot; alias Mr. Gervase Cary Elwes, sang an excellent topical song, attired in a quaint garb covered with interrogations, and carrying an electric telegraph-post in her hand. Lady Maud Warrender, as &quot;Pity,&quot; advanced from a barge that had just arrived, and sang a doleful ditty which made one wish &quot;Pity&quot; might combine a sense of gaiety. But as Mrs. Willie James, in the part of &quot;Mercy,&quot; dressed as a nurse, recited some bright lines anent Tommy, to the accompaniment of distant fifes and drums, the audience decided to take this as a satisfactory compensation.

All being now harmoniously arranged, &quot;War&quot; performed a sleight-of-hand feat, divested herself of her red dress, her headgear of flaming serpents, and her glistening breastpiece, and appeared in virgin white, crowned with roses, as “Peace,&quot; surrounded by “Music&quot; in a gorgeous gown of gold tissue, by “Painting,&quot; “Science,&quot; and “Literature.&quot; A pleasant finaleof gay music brought the Masque to a close, and left a decidedly agreeable and novel impression behind it.

Tableaux then followed, all more or less well grouped by well-known artists, and represented by beautiful women of Society. Among the familiar faces were Lady St. Oswald, Lady Mary Sackville, Miss Agatha Thynne, Mrs. Fitz Ponsonby, Lady Maitland, Madame von André, &amp;c., but neither Lady Helen Vincent, Lady De Grey, Lady Cynthia Graham, the Duchess of Portland, nor many other well-known and lovely ladies took part in the performance.

Finally, came the Patriotic Tableau, which had evidently engaged all the energies of the organisers of the fête. On a high throne, with a most realistic lion, open-mouthed and fierce-looking, beside her, sat Lady Westmoreland as &quot;Great Britain,&quot; a stately and dignified figure in white satin, draped in a red cloak and crowned with a large wreath of laurel. The stage on each side was lined by genuine stalwart Guardsmen, and to the sound of lively martial music, composed and conducted by Sir Arthur Sullivan, slowly advanced a procession of Great Britain's dependencies, figured by ladies magnificently costumed, their long jewelled trains borne by two little pages in cloth of gold brocade coats, with black silk legs. Very beautiful were the blendings of the colours in this tableau, artistically designed by Mr. Percy Anderson. Lady Claude Hamilton, as &quot;British Columbia,&quot; moved with stately gait in a robe of palest green; Lady Feo Sturt glittered barbarically with jewels; her headdress and her bosom were covered with gems. As the typical representative of &quot;India,&quot; she was dressed in apricot colour and bore branches of hibiscus in her hands. Mrs. Hwfa Williams, in blazing red, carried a parrot and some red flowers. The Hon. Barbara Lister looked lovely and picturesque in her violet robes under a massive wreath of wisteria blossoms; Lady Raincliffe, wearing a curious high head-dress, was dressed in white to represent &quot;Canada.&quot; &quot;Rhodesia&quot; made one of the prettiest figures in her khaki gown and cloak, with the coquettish hat and feathers and the red trimming associated with the Colonial Volunteers. &quot;Natal&quot; appeared appropriately clad all in black, while little &quot;Nigeria,&quot; for the nonce, wore spotless white robes. /

Miss Muriel Wilson spoke an ode, and looked striking in apricot and white, with a high diamond crown and a long standing-up white feather. None of the ladies suffered from shyness; they showed thorough acquaintance with the stage, and moved easily thereon. In fact, costumes, arrangements, music, and the glorious feast of beauty left nothing to be desired.

The final impression in one's mind was that the stage produces strange effects. It idealises some faces, hardens others, and alters many. The large wreaths, almost grotesque in size, proved eminently becoming, and the Grecian draperies carried away the palm for beauty. After them our modern dress seems stiff, angular, and inartistic. The whole performance was one to be commended, and will no doubt be as successful financially as it was from the aesthetic and spectacular point of view.

Mrs. James Stuart Wortley, who died last week, will be regretted by every class of society. This lady, a beauty in her youth, devoted the latter part of her life entirely to works of charity. She founded the East London Nursing Society, to the tender and skilful ministrations of which many a poor woman owes her return to health, and in every philanthropic scheme, emigration, the befriending of young servants, and the education of youth, she took a lively interest. Her clear sense, her logical grasp of subjects and her immense activity were of infinite service in everything she undertook, and her memory will smell sweet in the hearts of the many who loved and depended on her.

I really wonder at the patience of the British taxpayer. During the snow of this week Belgravia, Eaton, and other fashionable squares, remained a morass of slush, ice, and half-melted snow. The pavements as slippery as glass had not been cleansed, and only at the risk of one's life one made one's way from street to street. (Greville 7, Col. 1a-2a)&lt;/blockquote&gt;

'''25 January 1900, Thursday'''

David Lindsay, [[Social Victorians/People/Crawford and Balcarres|Lord Balcarres]] and Constance Lilian Pelly married:
&lt;blockquote&gt;
MARRIAGE OF LORD BALCARRES.

The marriage of Lord Balcarres, M.P. for North Lancashire, eldest son of the Earl of Crawford of Balcarres House, Fife, and Haigh Hall, Wigan, to Miss Pelly, daughter of the late Sir H. Peily, Bart., and granddaughter of the Earl of Wemyss, was solemnised yesterday (Thursday) at St Margaret's Church, Westminster, in the presence of a large gathering of friends.

Among the invited guests were the Earl and Countess of Crawford, the Dowager Countess of Crawford, the Earl of Wemyss, Lord and Lady Elcho, the Hon. E. Lindsay, the Hon. Lionel Lindsay, the Hon. Ronald Lindsay, Lord and Lady Cowper, Mr. A. J. Balfour, the Hon. L. Greville, and many othsrs. The service was fully choral, and was conducted by the Bishop of Stepney, assisted by the the Rev. Canon Gore. Mr Yorke, the stepfather of the bride, gave her away. She wore a dress of white velvet, draped with old Brussels lace, the gift of the Dowager Countess of Crawford: chiffon veil and wreath of natural orange blossoms. Her only ornament was a Maltese cross of diamonds, also the gift of the Dowager Countess of Crawford.

There were nine bridesmaids. Miss Pelly, sister of the bride) [sic], the Hon. Mary Vasey, the Hon. Cynthia Charteris, Miss Brodrick, Miss Sybil Brodrick, Miss Benita Pelly, the Hon. Aline Menjendie, Miss Daisy Benson, and Miss Madeline Bourke. They were attired alike in costumes of white de chine, with lace insertions, with blue chiffon hat, trimmed with plumes of white and blue ostrich feathers. They carried bouquets of violets, and wore red enamel brooches with diamond centres and pearl drops, the gifts of the bridegroom. The Hon E. Lindsay supported his brother as best man.

At the conclusion of the ceremony the guests drove to the town residence of the bride's mother in Queen Anne's Gate, where the wedding reception was held. Later in the day the newly-married couple left town for Wrest Park, Ampthill, kindly lent them for the honeymoon by Earl and Countess Cowper. Princess Louise (the Marchioness of Lorne) sent the bride a handsome silver basket as a wedding present.&lt;ref&gt;&quot;Marriage of Lord Balcarres.&quot; ''Dundee Courier'' 26 January 1900 Friday: 4 [of 8], Col. 6b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000164/19000126/105/0004.&lt;/ref&gt;
&lt;/blockquote&gt;

===February===
1900, February, a brief account of the Matherses' Isis ceremony appeared in &quot;the New York periodical the ''Humanist'', February 1900&quot; (Howe 201).

==== 15 February 1900, Thursday ====
A number of familiar people took part in [[Social Victorians/People/Arthur Stanley Wilson|Enid Wilson]]'s wedding to the Earl of Chesterfield:&lt;blockquote&gt;This afternoon, at St. Mark's Church, North Audley-street, the [[Social Victorians/People/Chesterfield|Earl of Chesterfield]] is to be married to Miss Enid Wilson, second daughter of Mr. Charles Wilson, M.P., of Marter Priory, Yorkshire, and 41, Grosvenor-square. The bride, who will be given away by her father, will wear a dress of white crêpe de chine wrought with silver and trimmed with fine old lace and ermin. The bridesmaids will be Miss Gladys Wilson, sister of the bride, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], her cousin, Lady Aldra Acheson, daughter of the Earl [[Social Victorians/People/Gosford|and Countess of Gosford]], Lady Mary Willoughby, daughter of the [[Social Victorians/People/Ancaster|Earl and Countess of Ancastor]], Lady Marjorie Carrington, daughter of [[Social Victorians/People/Carrington|Earl and Countess Carrington]], Miss Daphne Bourke, the four-year-old daughter of the [[Social Victorians/People/Bourke|Hon. Algernon and Mrs. Bourke]], [[Social Victorians/People/Balfour|Miss Balfour]], and [[Social Victorians/People/Paget Family|Miss Paget]]. Lace Empire dresses and long bright red cloth Directoire coats trimmed with sable and hats to match. They will carry sable muffs, the gifts to them of the bridegroom. Viscount Ednam, the [[Social Victorians/People/Dudley|Earl and Countess of Dudley]]'s only son, aged six, and Lord Wendover, the only son of Earl and Countess Carrington, aged something over four, will be the youthful trainbearers. After the wedding Mrs. Charles Wilson will hold a reception at 41, Grosvenor-square. The Prince of Wales has given Lord Chesterfield a remarkably handsome embossed silver cigarbox, lined with cedar, monogrammed and coroneted, accompanied by a letter written by his Royal Highness to the bridegroom, cordially wishing him every happiness. The Duke of Fife's gift is a cedar-lined plain silver cigarette-case.&lt;ref&gt;&quot;London Day by Day.&quot; ''Daily Telegraph'' 15 February 1900, Thursday: 8 [of 12], Col. 3b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001112/19000215/175/0008. Name in British Newspaper Archive: ''Daily Telegraph &amp; Courier'' (London). Print p. 8.&lt;/ref&gt;&lt;/blockquote&gt;Another, more local report: &lt;blockquote&gt;There was very large and fashionable assembly at St. Mark’s Church, North Audley-street, London, on Thursday afternoon, to witness the very pretty wedding of the Right Hon. the Earl of Chesterfield, P.C., of Holme Lacey, Hereford, and Miss Enid Wilson, fourth daughter of Mr. Charles H. Wilson, M.P. for Hull, of Warter Priory, York, and 41, Grosvenor-square, London. The service was fully choral, and the church handsomely decorated.

There were seven bridesmaids in attendance upon the bride. These young ladies were Miss Gladys Wilson (sister). Miss Muriel Wilson (cousin of the bride). Lady Mary Willoughby, Lady Alexandra Acheson, Lady Marjorie Carrington, Miss Dorothy Paget, and Miss Alice Balfour, who were in costumes of quaint, old-fashioned riding coats of red cloth, with white muslin skirts.

The local guests included Sir James and Lady Reckitt, Sir James and Lady Woodhouse, Lord and Lady Herries, Mr. Philip Hodgson, Lord and Lady Raincliffe, Mr. and Mrs. Kenneth Wilson, Mr. and Mrs. Stanley Wilson, Mr. and Mrs. Strickland Constable.

Presents wore also received from Commander Bethell (silver candlestick). Hr. and Mrs. George A. Duncombe, Beverley (Louis XV. sofa). Mr. and Mrs. Frank Lambert, Beverley (inlaid writing table), Colonel and Mrs. Goddard, Cottingham (silver mirror), Mr. Haslewood Taylor, Beverley (pair of prints). (1900-02-21 Beverley Echo)&lt;/blockquote&gt;

==== 27 February, 1900, Tuesday ====
Mardi Gras

===April 1900===

==== 8 April 1900, Sunday ====
Palm Sunday

====14 April 1900, Saturday====
Wynn Westcott assumed W. A. Ayton was on, as he wrote, &quot;the Committee to investigate the G. D. which contains Yeats, Bullock and I suppose Ayton&quot; (Howe 217).

====20 April 1900, Friday====
The R.R. et A.C. was code named Research and Archaeological Association (Howe 226)

====21 April 1900, Saturday====
The Inner Order of the Golden Dawn met at 116 Netherwood Road, West Kensington (Howe 227).

===May 1900===
====26 May 1900, Saturday====
Arthur Sullivan is visited by &quot;Sir George Martin, the organist at St. Paul's Cathedral, and Colonel Arthur Collins, one of the royal equerries&quot; to get him to write a Te Deum thanking God for the end of the Boer War (Ainger, Michael. Gilbert and Sullivan: a Dual Biography. P. 381.).

====30 May 1900, Wednesday====
Derby Day.

According to the Morning Post, &lt;quote&gt;The Derby Day. / The Archbishops of Canterbury and York hold a Reception of Colonial and Missionary Church Workers in the Great Hall of the Church House, 4.30 to 6.30. / ... May Fair and Bazaar, St. George's Drill Hall, Davies-street, Berkeley-square, opened by Lady Edward Spencer Churchill, 2.30.&lt;/quote&gt; (&quot;Arrangements for This Day.&quot; The Morning Post Wednesday, 30 May 1900: p. 7 [of 12], Col. 6C)

===June 1900===
Summer 1900: WBY summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

==== 3 June 1900, Sunday ====
Whit Sunday (Pentecost)

Whitsun party at Sandringham House, described by Lord Knutsford in his letters and summarized by Anita Leslie, whose parent's generation remembered some of these people Knutsford mentions as present:

* The Prince and Princess of Wales
* Princess Victoria
* Other daughters of the Prince and Princess of Wales
* Lord Knutsford
* [[Social Victorians/People/Ripon|Lord and Lady Gladys de Grey]]
* Luís De Soveral
* Tosti
* [[Social Victorians/People/Durham|Hon. George Lambton]]
* [[Social Victorians/People/Churchill|Lady Randolph Churchill]]
* [[Social Victorians/People/Holford|Holford]]
* Lady Musgrave

Leslie's summary of Knutsford's letters:&lt;blockquote&gt;The Whitsun party that year included Lord and Lady de Grey, De Soveral, whose caustic wit always lightened Edward's humour, Tosti, the famous baritone-songwriter (Alexandra and her daughters were so musical — strumming away ''à quatre mains'' while Totti's voice made chandeliers vibrate in after-dinner songs), the Hon. George Lambton (racing trainer), and Lady Randolph Churchill, &quot;just back from her hospital ship which had been a boon in South Africa, but fractiously insisting she is going to marry George Cornwallis-West.&quot;

Lord Knutsford describes the chattering guests travelling in that special train coach from St. Pancras to Wolverton Station where the house party was met by royal carriages with officious flunkeys in red livery who dealt with the luggage — and ''such'' luggage! Big trunks had to be brought for a few days' stay so that the correct attire could be produced for every meal and outing.

How exciting to drive through a forest of rhododendrons and to disembark in front of Sandringham House. The royal host and hostess stood in the hall to welcome their guests. After handshakes Queen Alexandra sat down to pour tea. Dinner was at 9 &lt;small&gt;P&lt;/small&gt;.&lt;small&gt;M&lt;/small&gt;. (at Sandringham all clocks were kept half an hour ahead of time). Footmen informed the gentlemen what waistcoats were to be worn. Ladies' maids scurried to the ironing rooms. At nine, having assembled in the drawing room, each man was told whom he must escort into dinner and where to sit. This saved hesitation and embarrassment. On this occasion Knutsford describes the Prince giving his arm to Lady de Grey, while Alexandra walked beside De Soveral and Lord de Grey escorted the unmarried Princess Victoria. There were, of course, no cocktails, but exquisite wines accompanied each course. The Prince never drank more than a glass or so of claret at dinner and a brandy after the last course.

When the ladies left the dining room cigarettes and cigars were brought by footmen. Heavy drinking was never encouraged, and / after half an hour the gentlemen moved to the drawing room to chat with the ladies, until Alexandra rose and they retired to their bedrooms where the ladies' maids would be waiting to unlace them from their gorgeous satin and velvet gowns. Hard as the existence of a servant might be, they were perhaps consoled by the colossal meals offered in recompense for late hours. A five-course breakfast could be consumed by every scullery maid if she so desired, and many a working-class mother strove to &quot;get her daughter's knees under a good table.&quot;

When the ladies had disappeared upstairs the men went to the billiards room, where the Prince, who idolised his dogs, would roar with laughter when his black bulldog nipped the legs of players. No one could go to bed before Edward, but at twelve-thirty he would certainly retire. There was no thought of any hanky-panky after hours at Sandringham. That would have been considered bad taste and an insult to the royal hostess.

On Sunday morning the breakfast gong sounded at 10 &lt;small&gt;A&lt;/small&gt;.&lt;small&gt;M&lt;/small&gt;. Then came church and a stroll in the garden until lunch at one-thirty. After a fairly heavy meal the ladies went upstairs to change into walking skirts and strong boots. The whole party then underwent a slow three-hour walk to the kennels and stables and farm. Talk was almost entirely about animals — dogs, pedigree cattle and, of course, race horses. Knutsford noticed Alexandra's &quot;touching girl-like love&quot; for every stone and corner of Sandrringham. She reminded him of &quot;a bird escaped from a cage.&quot;

Certainly the royal pair were never so happy as in this big Norfolk house, which they regarded as home, but guests grew weary of trying to do the right thing. Knutsford found dinner very wearing, with the conversation in mingled English and French: &quot;they drop from one to another in the same sentence.&quot;

Then came the local Whitsunday sports. Off drove the house party — Lady de Grey and Holford in the first carriage with Edward. Knutsford found himself in the second carriage with Princess Victoria and Lady Randolph Churchill and Lady Musgrave. The ladies wore coloured blouses and contrasting skirts and jackets over their blouses, white gloves and feather boas. A brisk wind nearly blew off their huge hats. Lady Musgrave in particular had difficulty with her concoction. &quot;Send it to the bazaar!&quot; cried Alexandra, and everyone roared with laughter.

Sandringham parties were called &quot;informal,&quot; but what a relief, nevertheless, when they all got back to the station in those regal carriages followed by the four horse-drawn vans of luggage.

In this spring of 1900 the visitors departed to their homes full to / the brim of food and anecdote. Jennie, who had been argumentative all weekend, would almost immediately marry her young George. Gladys de Grey would get on her newly installed phone to admirer number one, the Hon. Reginald Listen, or if he was not available to admirer number two, Sir John Listen-Kaye. Ladies were now able to ring the men up and guardedly converse instead of sending dangerous notes. Servants might overhear but there would be nothing ''on paper''.&lt;ref&gt;Leslie, Anita. ''The Marlborough House Set''. Doubleday, 1973.&lt;/ref&gt;{{rp|195–197}}&lt;/blockquote&gt;

====26 June 1900, Tuesday====
There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.

==== 28 June 1900, Thursday ====
Lady Randolph Churchill and George Cornwallis-West married at St. Paul's, Knightsbridge.&lt;ref&gt;Martin, Ralph G. ''Lady Randolph Churchill : A Biography''. Cardinal, 1974. Internet Archive: https://archive.org/details/ladyrandolphchur0002mart_w8p2/.&lt;/ref&gt;{{rp|220–223}}

===July 1900===
====27 July 1900, Friday====
The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] had dinner at the Arthur Wilsons’:&lt;blockquote&gt;[[Social Victorians/People/Arthur Stanley Wilson|Mr and Mrs Arthur Wilson]] were honoured with the presence of the Prince of Wales at dinner on Friday night. Amongst the guests were the Portuguese Minister, Count Mensdorff, Duke of Roxburghe, Lady Georgina Curzon, Captain and Lady Sarah Wilson (arrived that morning from South Africa), Lord and Lady Tweedmouth, Lord Herbert Vane Tempest, Viscount Villiers, Lady Norreys, Lady Gerard, [[Social Victorians/People/Keppel|Hon Mrs Keppel]], Sir Edward and Lady Colebrook, Mr and Mrs Grenfell, Lady Lister Kaye, Mrs Arthur Paget, Mr and Mrs Arthur Sassoon, Hon. W. Erskine, Mr and Mrs J. Menzies, General Oliphant, Miss Jane Thornewell, Mrs Kenneth Wilson, and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]]. (1900-07-30 Hull Daily Mail)&lt;/blockquote&gt;

===October 1900===
====31 October 1900, Wednesday====
Halloween.
===November 1900===
====5 November 1900, Monday====
Guy Fawkes Day
====9 November 1900, Friday====
A debutante dance for Miss Helyar: &lt;quote&gt;In honour of the coming of age of Miss Helyar, a small dance was given by Lady Savile, at Rufford Abbey, last night. The number of invitations was not so large as it would have been but for the war. The house party included Mrs. and Miss Cavendish Bentinck, Lady Juliet Lowther, Lady Evelyn Ward, Lady Mabel Crichton, Mrs Kenneth Wilson, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Sir Berkeley Sheffield, Miss Sheffield, Lord Hyde, Lord Herbert, the Hon. B. Ward, the Hon. E. FitzGerald, the Hon. W. Erskine, Mr. Laycock, Captain Brinton, the Hon. George Peel, Mr. Harris, Captain Tharp, Captain Heneage, and the Hon. G. Portman.&lt;/quote&gt; (1900-11-10 Yorkshire Post)

====27 November 1900, Tuesday====
Arthur Sullivan's funeral: &lt;quote&gt;At eleven o'clock on Tuesday, November 27th, the [366/367] funeral procession set forth from Victoria Street, Westminster, on its mournful way, first to the Chapel Royal, St. James's, where, by command of the Queen, part of the Burial Service was to take place, and thence to St. Paul's. Throughout the line of route flags drooped at half-mast, whilst beneath them people crowded in their thousands, bare-headed and in silence, waiting to pay their last tribute of respect and gratitude to the lamented master whose genius had done so much to brighten their lives for the past five-and-twenty years. [new paragraph] Into the Royal Chapel, where Arthur Sullivan had begun his career as a chorister, was borne the casket containing his remains. On either side stood men and women famous in society and the wider world of Art in all its branches. The Queen was represented by Sir Walter Parratt, Master of Music, who was the bearer of a wreath with the inscription: &quot;A mark of sincere admiration for his musical talents from Queen Victoria.&quot; Sir Hubert Parry represented the Prince of Wales; the German Emperor was represented by Prince Lynar, Attache of the German Embassy; Prince and Princess Christian by Colonel the Hon. Charles Eliot, and the Duke of Cambridge by General Bateson.

Among the congregation at the Chapel Royal were seen the United States Ambassador; the Earl and Countess of Strafford; Theresa, Countess of Shrewsbury; the Countess of Essex; Lord Glenesk; Lord Rowton; Lord Crofton; Lady Catherine Coke; the Dean of Westminster; Lady Bancroft; Lady [367/368] Barnby; Mr. Arthur Chappell; Mr. and Mrs. F. C. Burnand; Mr. Arthur W. Pinero; Mr. Haddon Chambers; Lieutenant Dan Godfrey; Signor Tosti; Mr. George Grossmith; Mr. Rutland Barrington; Miss Macintyre; Mrs. Ronalds; Canon Duckworth; Lady Lewis; Miss Ella Russell; Mr. Augustus Manns; Mr. Charles Wyndham; Captain Basil Hood; the Chairman and Secretary of Leeds Musical Festival; and Representatives of various British Musical Associations.

The Pall-bearers were Sir Squire Bancroft, Mr. Francois Cellier, Colonel A. Collins (one of the Royal Equerries), Sir Frederick Bridge, Sir George Lewis, Sir Alexander Mackenzie, Sir George Martin, and Sir John Stainer. [new paragraph] he chief mourners were Mr. Herbert Sullivan (nephew), Mr. John Sullivan (uncle), Mrs. Holmes, and Miss Jane Sullivan (nieces), Mr. Wilfred Bendall (Sullivan's secretary), Mr. B. W. Findon, Mr. Edward Dicey, Mr. C. W. Mathews, Mrs. D'Oyly Carte, Dr. Buxton Browne, Mr. Arthur Wagg, Mr. Fred Walker, Mr. Dreseden and Sir Arthur's servants. [new paragraph] Much to their regret, neither Mr. Gilbert nor Mr. Carte was able to attend the funeral. The first was on the Continent for the benefit of his health, the second was laid up by serious illness. The present writer also, having been absent from London at the time, has not the advantage of an eye-witness to give a graphic description of the funeral obsequies of his old friend; and so, rather than attempt to paint the picture from imagination, he gladly avails himself [368/369] again of the courtesy of his brother-author who is so generous as to lend the aid of his experience. [new paragraph] In these sympathetic words, Mr. Findon describes the scenes and incidents in which, as a chief mourner, he took part at the Chapel Royal and St. Paul's Cathedral: &lt;blockquote&gt;&quot;. . . As the casket was borne into the Chapel, it was impossible to avoid thinking of those days when Sullivan himself had worn the gold and scarlet coat of a Chapel Royal Chorister, and his sweet young voice had rung through the sacred edifice. Then the world and its honours lay before him, but we doubt if even in the most sanguine moments of impulsive boyhood he imagined the greatness that one day would be his, or that his bier would pass within those honoured walls amid the silent demonstration of a mourning people. The anthem, 'Yea, though I walk through the valley of the shadow of death,' from his oratorio 'The Light of the World,' was beautifully sung, and the pathos of the music bathed many a face in tears, and touched a tender spot in more than one loving heart. Another of the dead master's exquisite thoughts, ' Wreaths for our graves the Lord has given,' brought the Service at the Chapel Royal to an end, and the procession passed on its way to St. Paul's Cathedral, which was crowded with sympathetic spectators.

&quot;Clerical etiquette and cathedral dignity compelled the beginning of the Burial Service anew, and when the coffin had been lowered into the crypt there came the most poignant moment of the long ceremonial. [new paragraph] &quot;Close to the open vault sat the members of the Savoy Opera Company, including his life-long friend, Mr. Francois Cellier, who had been associated as chef d'orchestre with all his comic operas, and, after [369/370] the Benediction had been given, they sang in voices charged with emotion the touching chorus, 'Brother, thou art gone before us,' from ' The Martyr of Antioch.' The effect was quite remarkable, inasmuch as it was one of those incidents which come but rarely in a life-time.&quot;&lt;/blockquote&gt;It was not in London alone that people mourned for Arthur Sullivan on that November day. Throughout Great Britain and Ireland, on the Continent of Europe, in America and farther across the seas, thousands of fond and grateful hearts ached with grief at the thought that England's dear master of melody had passed away into the silent land. From high-born personages and from people of low estate came floral emblems, wreaths, crosses, and lyres innumerable. Conspicuous among them was a beautiful harp of purple blossoms with strings — one broken — of white violets. To this offering was attached a card bearing the inscription:&lt;blockquote&gt;In Memoriam

ARTHUR SEYMOUR SULLIVAN

Born 13 May, 1842. Died 22 Nov., 1900

FROM MR. D'OYLY CARTE'S &quot;ROSE OF PERSIA&quot; TOURING COMPANY IN TOKEN OF THEIR AFFECTIONATE REGARD
&lt;poem&gt;Dear Master, since thy magic harp is broken,
Where shall we find new melodies^ to sing?
The grief we feel may not in words be spoken;
Our voices with thy songs now heav'nward wing.
Whilst on thy tomb we lay this humble token
Of love which to thy memory shall cling.&lt;/poem&gt;
BELFAST,
24th November, 1900.&lt;/blockquote&gt; [370/371] These simple lines but half expressed the love and esteem in which Sir Arthur Sullivan was held by all whose privilege it was to have been associated with him, and to have served, however humbly, his proud and brilliant life-cause.&lt;/quote&gt; (Cellier, François, and Cunningham Bridgeman. Gilbert and Sullivan and their operas: with recollections and anecdotes of D. Pp. 366-371. Google Books: http://books.google.com/books?id=Au05AAAAIAAJ.)

====30 November 1900, Friday====
The wedding between Lady Randolph Churchill and George Cornwallis West at St. Paul's, Knightsbridge, occurred about this time. [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] attended, as did much of Society (1900-07-30 Times).

===December 1900===
===25 December 1900, Tuesday====
Christmas Day
====26 December 1900, Wednesday====
Boxing Day
===Works Cited===
*[1900-02-14 Morning Post] &quot;The Household Troops. Entertainment at Her Majesty's.&quot; Morning Post 14 February 1900, Wednesday: 3 [of 10], Col. 1a–2b [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19000214/014/0003 (accessed February 2020).
*[1900-07-30 Hull Daily Mail] &quot;Social Record.&quot; Hull Daily Mail 30 July 1900, Monday: 2 [of 6], Col. 5a [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000324/19000730/007/0002 (accessed July 2019).
*[1900-07-30 Times] &quot;Court Circular.&quot; Times, 30 July 1900, p. 6. The Times Digital Archive, http://tinyurl.galegroup.com/tinyurl/AHR8r5. Accessed 20 June 2019.
*[1900-11-10 Yorkshire Post] &quot;Court and Personal.&quot; Yorkshire Post 10 November 1900, Saturday: 6 [of 14], Col. 4c [of 8]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000687/19001110/099/0006 (accessed July 2019).
*Greville, Lady Violet. &quot;Place aux Dames.&quot; The Graphic 17 February 1900, Saturday: 7 [of 40], Col.1a–2a, 2c [of 3]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/19000217/008/0007 (accessed July 2019). [Col. 2c only for the last 2 paragraphs, not really relevant to Muriel Wilson]

==1901==
===January===
&quot;There were no winter performances of opera at Covent Garden in those times: there was, in 1901, only a summer season&quot; (Baring-Gould II 704, n. 14, quoting Rolfe Boswell).
====1 January 1901, Tuesday, New Year's Day====
====16 January 1901, Wednesday====
Arnold Dolmetsch sent out notices that he was moving to 85 Charlotte Street, Fitzroy Square (Campbell 137-38).

====22 January 1901, Tuesday====
Queen Victoria died at Osborne House, on the Isle of Wight.

====23 January 1901, Wednesday====
Edward VII formally proclaimed “King of Great Britain and Ireland and Emperor of India, Defender of the Faith” &quot;at Temple Bar, on St. Paul's Cathedral steps and at the Royal Exchange.&quot; &quot;The Privy Council met in St. James' Palace at 2 o'clock in the afternoon for the purpose of signing the accession proclamation of Edward VII. The attendance at the meeting of the Council was more than 200.&quot; (Merrill, Arthur Lawrence, and Henry Davenport Northrop. Life and Times of Queen Victoria: Containing a Full Account of the Most Illustrious Reign of Any Soveriegn in the History of the World, Including the Early Life of Victoria; Her Accession to the Throne and Coronation; Marriage to Prince Albert; Great Events During Her Brilliant Reign; Personal Traits and Characteristics That Endeared Her to Her People; Graphic Descriptions of Her Charming Home Life; Noble Qualities as Wife and Mother; Royal Castles; Public Receptions; Wonderful Growth of the British Empire, Etc. To Which is Added the Life of King Edward VII., and Sketches of the Members of the Royal Family. Philadelphia, PA: World Bible House, 1901. Page 437. Google Books: http://books.google.com/books?id=Kx48AQAAIAAJ)

====26 January 1901, Saturday====
Arnold Dolmetsch gave a performance at his new domicile at 85 Charlotte Street, Fitzroy Square (Campbell 137-38).

===February 1901===
====2 February 1901, Saturday====
Queen Victoria’s funeral at St. George’s Chapel, Windsor Chapel. Consuelo (Vanderbilt), Duchess of Marlborough was there: &lt;blockquote&gt;The service itself was magnificent. The stalls of the Knights of the Garter were occupied by the German Emperor and a dazzling array of kings, queens, ambassadors extraordinary, Indian princes, Colonial dignitaries, generals, admirals and courtiers. Consuelo wore the prescribed deep black mourning and crepe veil, which rather suited her, and it had the effect of extracting what she describes as a 'rare compliment' from her husband who remarked: 'If I die, I see you will not remain a widow long' — a conceit which suggests that he was more of his father's son than he cared to acknowledge.

Consuelo later reflected that the funeral of Queen Victoria was a moment when it truly appeared that no other country in the world had an aristocrac so magnificent, nor a civil service so dedicated, which is precisely what was intended. The great doors were flung open as the royal cortege mounted the steps, a boom of distant guns and clanging swords the only sound other than the funeral march, until Margot Asquith broke the reverential silence with a quip. Consuelo thoroughly enjoyed herself at the reception in the Waterloo Chamber afterwards too. (Stuart, Amanda Mackenzie. Consuelo and Alva Vanderbilt: The Story of a Daughter and a Mother in the Gilded Age. New York and London: HarperCollins, 1005. Page 228. Google Books: http://books.google.com/books?id=44mhoIv12rEC)&lt;/blockquote&gt; Also Henry James saw the funeral procession.

====3 February 1901, Sunday====
1901 February 2–4?: Queen Victoria lay in state for 2 days between her funeral and her interment.

====4 February 1901, Monday====
Queen Victoria’s interment at Frogmore Mausoleum, Windsor Great Park.
====23 February 1901, Saturday====
The wedding of Hugh Richard Arthur, 2nd Duke of Westminster and Constance Edwina Cornwallis-West (1901-02-23 Cheshire Observer).

===March 1901===
Sometime in March 1901 Arthur Conan Doyle and Fletcher Robinson &quot;were on a golfing holiday at the Royal Links Hotel at Cromer in Norfolk,&quot; where Robinson told Doyle a Dartmoor legend of &quot;a spectral hound&quot; (Baring-Gould II 113).

Doyle's &quot;The Hound of the Baskervilles&quot; began publication in the ''Strand'' in January 1902.

===April 1901===
====18-20 April 1901, Thursday-Saturday====
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] and Mrs. Beerbohm Tree took part in 3 performances of &lt;quote&gt;Masks and Faces. The matinées have been organized by [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]], of Tranby Croft, in aid of the local fund of the Soldiers’ and Sailors’ Families Association. It was originally intended that the matinées should have been given in January last, but, owing to the death of Queen Victoria, they were postponed until Thursday, Friday, and Saturday last week. Additional interest was centered in the event, owing to the cast including no less a name than that of Mrs. Beerbohm Tree, while the fact that Miss Muriel Wilson was to appear as Peg Woffington aroused expectation.&lt;/quote&gt; (1901-04-25 Stage)

===May 1901===
'''1901 May 30, Thursday''', the Ladies' Dog Show opened:&lt;blockquote&gt;Yesterday the annual show of the Ladies' Kennel Association was held in the Royal Botanical Gardens, Regent's Park, and attracted a highly fashionable gathering. Among the ladies represented were Princess Victor Dhuleep Singh, Princess Sophie Dhuleep Singh, the Marchioness of Nottingham, the Duchess of Sutherland, the Countess of Aberdeen, Lady Evelyn Ewart, Lady Helen Forbes, the Hon. Mrs. Baillie, Lady Moor, the Hon. Mrs. Alwyne Greville, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], Lady Alwyne Compton, Lady Chetwode, Lady Cathcart, Lady Angela Forbes, the Hon. Mrs. Fellowes, Lady Gooch, Princess de Montglyon, and Viscountess Southwell, Mrs. Samuelson, Miss Serena, Mrs. Bosanquet, Mrs. Williams, and Mrs. Ingle Bepler. Cats and poultry are also exhibited.&lt;ref&gt;&quot;Ladies' Dog Show.&quot; ''Birmingham Daily Gazette'' 31 May 1901, Friday: 6 [of 8], Col. 5b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000667/19010531/124/0006. Print p. 6.&lt;/ref&gt;&lt;/blockquote&gt;

===June 1901===
Summer 1901: William B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).
====17 June 1901, Monday====
&lt;quote&gt;The &quot;Women Writers&quot; held their dinner at the Criterion on Monday, the 17th. Now Mr. Stephen Gwynn, in his paper entitled &quot;A Theory of Talk,&quot; roundly asserts that women are less amusing than men. He says that there is no reason in nature why they should be, but that their inferiority is obvious. He points out that &quot;thirty or forty men will meet at seven o'clock, dine together, and pass the evening very agreeably till midnight. Imagine thirty or forty women called upon to do the same; would they be able to amuse themselves?&quot; It seems almost a pity that the exclusiveness of the women writers would not allow Mr. Gwynn personally to observe whether they were amused or bored on Monday night. In number there were nearly two hundred, and there certainly did not appear to be any lack of enjoyment or of laughter, but then it is also a fundamental belief with men that women are early adepts at hiding their true feelings. / Lucas Malet occupied the chair, and her carefully prepared speech was read out by Miss Sydney Phelps. Standing at the base of the statue of one of the world's greatest authors, and that, we regret to say, not a woman but a &quot;mere man,&quot; Miss Phelps, speaking for Lucas Malet, said there was good cause for women to congratulate themselves that, whereas there had been Thackeray, Dickens, the brothers Kingsley, and Wilkie Collins among authors, authoresses could boast of George Eliot, Mrs. Gaskell, [33 Col B / 34 Col A] Miss Yonge, &amp;c, and she felt that in the future they might equal, she would not say rival, their &quot;brother man.&quot; At this courageous vaunt our glance involuntarily strayed to the statue, anticipating that it would be moved to at least a wink; but overwhelmed, perhaps, by the presence of so many &quot;sisterwomen,&quot; it gave no sign. The speech was long, lasting for over thirty minutes. It touched on the evils of lowering work to what might be a present commercial but fleeting value; it contained much that was excellent, and tendered some good sound advice; perhaps it dwelt a trifle too insistently upon the obvious, and it was serious even to solemnity. But then &quot;women are so serious.&quot; / Mme. Sarah Grand's reply was couched in far lighter vein. It slipped into the anecdotal, and was altogether more in the masculine line of after-dinner speaking. It offered no advice save on the advisability of laughter; it lingered for a moment on the sorrows of misinterpretation and misunderstanding, and included some amusing examples. Mme. Sarah Grand possesses a sympathetic voice, and is very pleasant to listen to. / It is characteristic of the gravity with which even in play hours women regard their &quot;work&quot; that the majority of guests preferred the more serious matter of Lucas Malet to the light personal note of Mme. Grand. The dinner itself was very good, and it was noticeable that whilst at the Authors' dinner on May 1 but few women availed themselves of the permission to smoke, at the women's function scarcely one was without a cigarette. Coffee was served at the table, and afterwards the company broke up into groups. / The committee numbered among its members Miss Beatrice Harraden, Mrs. Steel, Mrs. Craigie, Miss Christabel Coleridge, Miss Violet Hunt, and many other favourite writers. In the company present there were Dr. Jex-Blake, Mrs. Ady, Dr. Margaret Todd, Miss Adeline Sergeant, Mrs. Mona Caird, Mrs. Burnett-Smith, Mme. Albanesi, Miss Nora Maris, Miss Kenealy, and others; and the following presided at the tables : Lucas Malet, Mme. Sarah Grand, Mrs. de la Pasture, Miss Montresor, the Lady Mayoress, Mrs. L. T. Meade, Mrs. Alec Tweedie, Mrs. Walford, Mrs. B. M. Croker, Miss Violet Hunt, Miss Beatrice Harraden, Mrs. Belloc Lowndes, Miss Violet Brooke-Hunt, Miss Thorneycroft Fowler.&lt;/quote&gt; (&quot;The Women Writers' Dinner.&quot; The Author. Vol. XII, No. 2. 1 July 1901. Pp. 33–34.)

====26 June 1901, Wednesday====
There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.

====29 June 1901, Saturday====
&quot;To-day sees the public inauguration of the Horniman Musem at Forest Hill. This collection of marvels from many lands, gathered together by a member of the Horniman family, has been generously presented to the public and housed in a handsome new building — set in the midst of fifteen acres, which are now dedicated to use as a public park. The entrance to the museum will be free.&quot; (&quot;The Horniman Museum.&quot; Illustrated London News (London, England), Saturday, June 29, 1901; pg. 928; Issue 3245, Col. B)

===July 1901===
====19 July 1901, Friday====
[[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] hosted a concert at the Wilson house in Grosvenor-place in London: &lt;quote&gt;Mr. and Mrs. Arthur Wilson lent their house in Grosvenor-place on Friday afternoon for Miss Gwendoline Brogden’s concert. Miss Brogden, who is only eleven years old, is quite a prodigy. She sings quite exquisitely, and great many people, including Lady de Grey and Mrs. Arthur Wilson, are much interested in her future, which promises to be a very brilliant one. Lady Maud Warrender, Miss Rosamond Tufton, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. Bernard Ralt, Signor Ancona, and Signor Tosti, all promised to assist at the concert, and the tickets were a guinea each.&lt;/quote&gt; (1901-07-24 Beverley Echo)

==== 25 July 1901, Thursday, 2:30 p.m. ====
The wedding of William Dixon Mann Thomson — Captain Mann Thomson in the Life Guards — and Violet Hemsley Duncan. Captain Mann Thomson's father had died in 1899. (Guests' names with their gifts set as an unordered list here, to save space; it was typeset as a long list of paragraphs in the newspaper story.)&lt;blockquote&gt;MARRIAGE OF CAPTAIN MANN THOMSON AND MISS DUNCAN.

The marriage of Captain Mann Thomson, Royal Horse Guards, and Miss Violet Duncan, eldest daughter of Mr. A. Lauderdale Duncan, Knossington Grange, Oakham, took place in St. Peter's Chnrch, Eaton-square, London, on Thursday, the inst., 2.30 p.m. The bride, who was given away her father, wore a dress of white satin, draped with white and old Brussels lace, wreath of orange blossoms, and tulle veil. Her ornaments were pearls. She was attended by seven bridesmaids, viz.: — Miss Adèle, Miss Marjory, and Miss Esmè Duncan, sisters; Miss Dorothy and Miss Sybil Thompson, cousins of the bride; Miss Villiers, cousin of the bridegroom; and Miss Joan Dawson. They wore dresses of the palest pink silk, covered with pink gauze, collars of white lace, and pale pink chiffon baby hats. The bride's train was carried by Miss Duncan, her youngest sister. The bridesmaids carried bouquets of pink carnations, and wore diamond brooches in the shape of a violet with sapphire centre, the gifts the bridegroom. A detachment of non-commissioned officers and men of the bridegroom's troop lined the aisle during the ceremony. The bridegroom was supported by the Earl Arran as best man. The officiating clergy were the Rev. Ravenscroft Stewart, Vicar of All Saints', Ennismore-gardens, the Rev. G. Tanner, Rector of St. Peter's, Knossington, Leicestershire, and the Rev. H. Trower. After the ceremony, a reception was held at 8, Rutland-gate, the residence of Mr. and Mrs. Lauderdale Duncan. Among those present were the Duke and Duchess of Westminster, Dowager Countess of Chesterfield, Sir William and Lady Houldsworth, the Hon. C. and Mrs. Stanhope, Miss Hay, Lord and Lady Eglinton, Lord and Lady Castlereagh, Lord Ernest St. Maur, Lord and Lady Pembroke, Mrs. Adair, Mrs. Mann Thomson, Miss Mann Thompson, Earl Arran, Lord Cecil Manners, Mrs. and Miss Wilton Phipps, and many others. Later, the bride and bridegroom left for Dover, ''en route'' for the Continent, where they will spend the honeymoon. The bride's travelling dress was of pale blue crepe-de-chine, and black hat. There were about five hundred gifts from relations and friends. The following is a list:—

* Bridegroom to Bride — Large diamond spray
* Mrs. Mann Thomson (mother of bridegroom) — Diamond ring, diamond and sapphire bangle, and cheque
* Mr. Lauderdale (father of bride) — Diamond and sapphire necklace
* Mrs. Duncan (mother of bride) — Silver-mounted travelling bag
* Dowager Lady Hay (bride's aunt) — Silver tea service
* Miss Mann Thomson (bridegroom's sister) — Brougham
* Mr. and Mrs. Butler Duncan (uncle and aunt) — Gold-mounted claret jug
* The Misses Jackson (bridegroom's aunts) — Silver plate
* Mr. H. Mann Thomson (brother) — Silver-mounted portmanteau
* Mr. Charles Hunt — Diamond and pearl brooch
* Miss Adele Duncan — Gold match-box
* The Earl Arran — Gold cigarette case
* Mr. and Mrs. Lucas — Bracelet
* Earl of Arran — Set of diamond and pearl studs
* Capt. and Lady Riddell — Bracelet
* Mrs. and Miss Wilton Phipps — Gold and ruby buckle
* Hon. H. Stanhope, R.N. — Brilliant buckle
* Mr. and Mrs. Pennington — Ruby necklace
* Mr. A. Butler Duncan — Necklace (old design)
* Mr. and Mrs. Gervase Beckett — Sleeve links
* Duke and Duchess of Westminster—Pair of silver candlesticks
* Duchess of Roxburgh—Dresden china coffee service
* The Countess of Shaftesbury — Walking-stick
* The Earl of Arran — Umbrella
* Lady Napier Magdala — Snuff-box
* Sir Richard Waldie Griffith — Fan
* Officers of the Royal Horse Guards — Massive silver vase
* Lady Houldsworth — Silver inkstand
* Viscount Ingestre — Silver waiter
* Miss Hay — Silver coffee pot
* Lady Hay — Silver tea caddy
* The Countess of Chesterfield — Silver and brilliant-mounted photo frame
* Lord Manners — Set four silver candlesticks
* Lord and Lady Eglinton — Silver cigarette box
* Earl and Countess of Ancaster — Pair of silver peppers
* Lady Augusta Noel — Book-slide
* Mr. and Mrs. Bradley-Martin — Old china coffee service in case
* Mr. and Lady Wilfred Renshaw — Leather-covered book, &quot;Where It?&quot;
* Mrs. Duncan — Silver-mounted stationery case and blotter
* Sir Arthur Fludyer — Hunting crop
* Lady Katherine Cole — Walking-stick
* Lord Hamilton — Oak card table
* Sir John Kelk — Writing case
* Capt. Hon. E. St. Aubyn — Set of silver spoons in case
* Capt. and Mrs. Burns-Hartopp — Set of silver asparagus tongs in case
* Capt. Trotter — Silver sealing-wax stand
* Capt. E. W. Clowes — Silver tobacco box
* Mr. and Mrs. Sands Clayton — Silver scent bottle
* Mr. and Mrs. John Hunt Clayton — Thermometer in silver-mounted case
* Mr. and Mrs. Evan Hanbury — Clock
* Major Atherley — Cigarette box
* Mr. and Mrs. Richard Tryon — Card case
* Mr. and Mrs. Hamilton Stubber — Table mirror in silver frame
* Mr. and Mrs. Gretton — Pair of silver candlesticks
* Miss Adele Duncan — Silver tea service
* Hon. G. Crichton — Silver-mounted paper-knife
* Mrs. Norman Lampson — Parasol
* Capt. Gregson — Photo, &quot;Guards at Pretoria&quot;
* Mr. Alfred Keyser — Leather bag
* Mr. and Mrs. Armytage — lvory paper knife
* Mrs. Boyce — Leather tray with two painted china plaques
* Mr. and Mrs. A. B. Norman — Silver-mounted paper knife
* The Master of Elibank — Pair of silver ash trays
* Mr. Adrian Rose — Pair of silver toast racks
* Mr. Archibald Smith — Hunting crop
* Major Bradford Atkinson — Walking-stick
* Mr. and Mrs. Stanhope — Painted china tea service
* Mr. G. A. Grant — Stationery case
* Mrs. Charles Inge — Copper and brass jardiniere
* Col. and Mrs. Makins — Hunting crop
* Mr. G. F. Trotter — Walking stick
* Mr. and Misses Cardwell — Fan
* Mrs. Dana — Thermometer
* Mrs. Nugent — Card case
* Mr. and Mrs. Ovey — Tortoiseshell box
* Mr. F. Peake — Writing table
* Capt. Boyce — Embroidered table cover
* Mrs. Duncan — Dressing bag case
* Mr. F. C. Fardell and Miss Gilbert Day — Brocaded satin cushion
* Mr. and Mrs. Niel Robson — Visiting book
* Mrs. R. B. Hay — Silver salts in case
* Mr. and Mrs. Harold Broadbent — Pair silver peppers in case
* —— Set silver knives in case
* Mr. and Mrs. Greville Clayton — Six silver vases in case
* Mr. and Mrs. Reginald H. Lewis — Pair silver peppers
* Lord Ernest St. Maur — Set four silver fruit spoons in case
* Rev. Geo. and Mrs. Tanner — Pair of silver salts
* Capt. Thomson's Valet and Groom — Pair of silver peppers
* Mr. Alick Duncan — Silver jug
* Mr. and Mrs. A. Brocklehurst — Silver timepiece in case
* Lieut.-Col. and Mrs. Blackburn — Silver fruit spoon
* Mr. and Lady Georgiana Mure — Silver-mounted ink [sic]
* Mrs. Gerald Fitzgerald — Silver-mounted inkstand
* Mrs. Ruthven — Set of silver knives in case
* Mrs. Blair — Umbrella
* Mrs. Willie Lawson — Hunting crop
* —— Three driving whips
* —— Tea tray
* Mr. and Mrs. Ramsay — Umbrella
* Mr. George Hunt — Silver flower bowl
* Mr. and Mrs. Reginald Cookson — Silver biscuit box
* Mr. Arthur and V. James — Silver two-handled cup and cover
* Mr. Robbio Stubber — Pair of silver scent bottles
* Mr. and Mrs. Geo. Baird — Silver bowl
* Mr. and Mrs. Harrison Broadley — Pair of silver flower vases
* Mrs. Grant—Silver flower-pot stand
* Mrs. Villiers — Silver corkscrew
* Capt. Spender Clay — Antique silver snuffbox
* Mr. and Mrs. Weir — Silver bacon dish
* Mr. Baird — Pair of silver candlesticks
* Mr. Athol Hay — Silver sugar bowl
* Capt. Ewing — Pair of silver fruit dishes
* Mr. and Mrs. C. J. Phillips — Pair of silver baskets
* Miss Esmé Duncan — Silver box
* Mr. and Mrs. Ronald Paton — lvory paper knife
* Dr. Freshfleld — Work case
* Mrs. Arkwright — Silver-mounted blotter
* Mr. and Mrs. Peake — Silver-mounted stationery case
* Miss Goddard — Book
* Mr. D. Baird — Silver inkstand
* J. G. and Jane B. Hay — lnkpot, with silver watch top
* Mr. and Mrs. Wadsworth Ritchie — Pair of silver dishes in case
* Mr. and Mrs. Guy Fenwick — Set of twelve silver knives in case
* Jane and Uncle Willie — Silver sugar basin in case
* Mr. and Miss Millington Knowles — Set of four silver dessert spoons in ease
* Herbert and Lady Beatrix Herbert — Silver flower dish
* Mr. and Mrs. J. B. Thorneycroft — Four silver candlesticks
* Mr. and Mrs. Russell? M [illegible, ink has spread] — Silver bowl [Col. 2c / Col. 3a]
* Mr., Mrs., and the Misses Wm. Cooper — Fan
* Miss Winearls — Silver-mounted scent bottle
* Sir Ernest Cassel — Diamond and enamel brooch
* Mr. John S. Cavendish — Gold pencil case
* —— Diamond and sapphire bracelet
* Miss Lottie Coats — Diamond and pearl brooch
* Hon. T. Robarts — Diamond brooch
* Mr. and Mrs. Chas. E. Hay — Enamel and pearl miniature holder
* Evelyn Ward — Cornomandel [sic] box
* Mr. and Mrs. Slade — China clock
* Lieut.-Col. Jervoise — Fan
* Mr. and Mrs. J. B. Fergusson—Set of four silver menu holders
* Mr. Guy R. F. Dawson — Silver card case
* Rev. E. V. and Mrs. Hodge — Silver dish
* Mr. C. S. and Mrs. Newton — Silver waiter
* Mrs. Metcalfe — Gold, turquoise, and ruby brooch
* Lord and Lady Erne — Set of three gilt decorated liqueur decanters
* Mr. and Mrs. Chas. Grant — Two silver-mounted spirit decanters
* Mr. and Mrs. George Baird — Set of three cut-glass decanters
* Mr. Peter Cookson—Pair of silver-mounted decanters
* Mrs. Featherstonehaugh — China ornament
* Aunt Mary — China coffee service in case
* Mr. H. S. Sykes — Silver-mounted telegram form case
* Capt. Meade — Pair of engraved claret jugs
* Lord and Lady Binning — Silver-mounted claret jug
* Mr. and Mrs. Baldock — Silver-mounted water jug, with inscription
* Mrs. and the Misses Chaplin — Pair of gilt decorated vases
* —— Silver-mounted claret jug
* Kittie, Margie, Hestie, Walter, Phillip, and Millicent Tanner — Pair of silver peppers case
* Mr. J. R. J. Logan — Silver-mounted claret jug
* Miss Ethel Baird — Painted china box
* Mrs. D. A. Neilson — Pair of female figures with Cupids
* M. M. Phillips — Painted china miniature box
* Lady Waldie Griffith — Stationery case
* —— Painted two-fold screen
* Miss Mabel Fitzgerald — Silver-mounted vase
* Major Bouverie — Silver-mounted match holder
* —— Enamelled inkstand and candlesticks to match
* Mrs. Duncan — Stationery case and blotter
* —— Silver-mounted stationery case
* —— Tortoiseshell and silver-mounted paper-knife
* Miss Mills — Dresden china vase, cover, and stand
* —— Six Vols. of Ruskin's &quot;Modern Painters&quot;
* Mrs. W. Baird — Leather bag
* Miss Langridge — Four silver spoons
* Miss Kirk and Miss Hemsley — Silver-mounted photo frame
* Miss Nessie Hemsley — Silver-mounted photo frame
* Captain and Mrs. St. Aubyn Loftus — Silver vase
* Decima Walker Leigh — Pair of silver-mounted menu stands
* Mrs. Charles Thomson — Mirror in silver frame
* Miss Reese — Silver crumb scoop
* —— Silver-mounted seal and case
* Mary Abercorn Alexander and Gladys Hamilton — Silver inkstand
* Mr. and Mrs. Cecil Chaplin — Silver pen, pencil, and knife in case
* Miss Gwendoline Brassey — Silver-mounted ice pail
* Mr. and Mrs. and Misses Clifford Chaplin — Pair of silver candlesticks
* Mr. and Mrs. Magee — lvory paper knife
* Misses Dorothy and Maude Pilcher — Scent bottle
* Miss Ashton — Silver-mounted clock
* Mrs. William Clarence and Miss Watson — Silver crumb scoop
* Major and Mrs. Ed. Baird — Egg-boiler on silver stand
* Mr. A. F. H. Fergusson — Pair of silver coffee pots
* —— Table mirror
* —— Pair of silver vases
* Mrs. R. B. Mnir — Silver fox ornament
* Mr. H. Brassey and Mr. H. R. Molynenx — Silver teapot
* —— Pair of silver sauce boats
* Mr. and Mrs. Heathcote — Silver cream jug
* Misses Thompson — Silver photo frame
* Mr. C. D. Rose — Pair of silver fruit dishes
* Mr. T. Archibald Hope — Silver toast-rack
* Mr. and Mrs. Robert Hunt — Pair of silver sauce boats
* Major and Mrs. Candy — Pair of silver fruit baskets
* Misses Trefusis — Silver-mounted owl mustard-pot
* Mrs. Frank Chaplin — Silver photo frame
* Major Vaughan Lee — Silver waiter
* Major Byng — Pair of silver menu stands
* Lady Wilton — Silver photo stand
* Geoffrey and Sibyll Palmer — Scent bottle
* Dr. Clement Godson — Silver salad cruet
* Mr. Mackenzie — Silver cigar case
* Mr. G. Colvin White — Set of four silver trays
* Mr. Edgar Brassey — Silver pipe lighter
* Miss Emily Dawson — Photo frame
* Mrs. Gerald FitzGerald — Silver match-box holder
* A. Barns — Silver waiter
* Miss Palmer — Letter-clip and dish
* Mr. and Mrs. Aubrey Coventry — Photo frame
* —— Silver bowl three feet
* Mr. and Mrs. Hornsby — Openwork silver basket
* —— Antique silver box
* Mr. and Mrs. H. R. Baird — Silver coffee-pot
* —— Pair of silver salts
* Mr. Hugh Wanemley — Silver-gilt match-box
* Captain Gordon Wilson — Silver snuff-box
* Mrs. Whitelaw — Silver mustard-pot
* Mrs. Palmer — Silver spoon
* Mr. Dudley Majoribanks — Silver bowl and cover
* Mr. Wilfred F. Ricardo — Pair silver candlesticks
* Indoor Servants at Knossington Grange and 8, Rutland Gate — Breakfast warmer and two silver entree dishes and covers
* Outdoor Servants at Knossington Grange — Silver stationery case
* Mr. Waterman (coachman) — Driving-whip
* Mr. Alexander (coachman) and Mrs. Alexander — lnk-stand
* Villagers of Knossington — Silver sugar bowl, sugar tongs, and cream ewer in case
* Silver vase, with inscription — &quot;Capt. Mann Thomson, Royal Horse Guards, from the Estate and Household at Dalkeith, on the occasion of his marriage, 25th July, 1901.&quot;
* Miss Baldock — Pair of scent bottles
* Captain Cook — Paper-knife
* Sir A. Baird — Pair of silver muffineers
* Rev. H. W. Trower — Pair of silver peppers
* Mr. T. Vandeleur — Silver cigarette box
* Lady Miller — Silver milk jug
* Mr. Hedworth Barclay — Silver muffineer
* Miss May A. Jackson — Photo frame
* Mr. Geoffrey Heneage — Silver ash tray
* Mr. and Mrs. R. B. Hay — Pair silver mustard-pots
* Mrs. George Charteris — Silver-mounted calendar
* Royal School of Art Needlework, Exhibition-road — Silvered copper heart-shaped box
* Mr. A. C. Newbigging — Silver fox ornament
* Mr. S. Schreiber — Silver match box
* Mr. and Mrs. J. H. J. Phillips — Silver muffineers
* Mr. and Mrs. Fyfe Jameson — Silver flask
* Mrs. Beaumont Lubbock — Silver bon-bon dish
* Lord Castlereagh — Salad bowl
* Captain Hambro — Silver card case
* Lord Longford — Silver bowl
* Captain —— Silver waiter
* Mrs. Forester — Silver frame
* Mrs. Martin — Tea cloth
* Mr. and Mrs. Cooper — Whip
* Earl Lonsdale — Silver tray
* Lady Augusta Fane — Red box
* Mr. Paul Phipps — Clippers
* Mr. E. Herlick — lnkstand&lt;ref&gt;&quot;Marriage of Captain Mann Thomson and Miss Duncan.&quot; ''Grantham Journal'' 27 July 1901 Saturday: 2 [of 8], Cols. 2a–3b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000400/19010727/003/0002.&lt;/ref&gt;
&lt;/blockquote&gt;

===August 1901===
====30 August 1901, Friday====
[[Social Victorians/People/Horos|The Horoses]] (troublesome members of the Golden Dawn) were thrown out of 99 Gower Street and moved to Gloucester Crescent (King 89 91).
===October 1901===
====31 October 1901, Thursday====
Halloween.
===November 1901===
====5 November 1901, Tuesday====
Guy Fawkes Day
===December 1901===
====25 December 1901, Wednesday====
Christmas Day
====26 December 1901, Thursday====
Boxing Day
===Works Cited===
*[1901-02-23 Cheshire Observer] &quot;Duke of Westminster. Brilliant Function.&quot; Cheshire Observer 23 February 2901, Saturday: 6 [of 8], Col. 1a–6c [of 8]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000157/19010223/114/0006 (accessed July 2019).
*[1901-04-25 Stage] &quot;Provinces.&quot; &quot;Amateurs.&quot; The Stage 25 April 1901, Thursday: 11 [of 24], Col. 3c, 4b–c [of 5]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001179/19010425/028/0011 (accessed July 2019).
*[1901-07-24 Beverley Echo] &quot;Stray Notes.&quot; Beverley Echo 24 July 1901, Wednesday: 2 [of 4], Col. 4b [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001561/19010724/037/0002 (accessed July 2019).

==1902==
Sometime in 1902, London publisher [[Social Victorians/People/Working in Publishing#George Newnes|George Newnes]] published an edition of [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s first (1892) collection of Holmes stories.
===January 1902===
====1 January 1902, Wednesday, New Year's Day====
===February 1902===

==== 13 February 1902, Thursday ====

King Edward VII and Queen Alexandra were present with some of their friends at Niagara, which must have been an ice-skating rink. Mr. and [[Social Victorians/People/Churchill|Mrs. George West]] are Lady Randolph Churchill and George Cornwallis-West.&lt;blockquote&gt;SOCIAL &amp; PERSONAL

Royalty at Niagara.

Quite a record audience was present at Niagara yesterday, when the free skating and waltzing competitions were skated off to the sound of gay music in a brightly lighted, warm atmosphere.

The royal box made a goodly show with its trappings of Oriental hangings and decorations of palms.

The Royal Box.

The King and Queen were accompanied by Princess Victoria and Prince and Princess Charles of Denmark, the Prince and Princess of Wales having previously arrived. Their Majesties were conducted to the spacious box by Mr. Hayes Fisher.

All the royal ladies wore black, the Queen adding a bunch of yellow Lent lilies to her sombre attire. Her two daughters lightened their mourning with touches of white, and the Princess of Wales wore a bunch of violets in her toque, with a twist of white.

In the adjoining box, among members of the suite were the Countess of Gosford, Earl Howe, Mr. Sidney Greville, Mr. H. J. Stonor, Lieut.-Colonel Davidson, Lieut.-Colonel Legge, and Viscount Crichton.

In boxes on the other side of the royal box were Lady Alice Stanley, with the Ladies Acheson, the Countess of Derby, Countess De Grey and Lady Juliet Lowther, [Col. 3c/4b] Mr. and [[Social Victorians/People/Churchill|Mrs. George West]] [Lady Randolph Churchill and George Cornwallis-West], Sir Edgar and Lady Helen Vincent, the Duchess of Bedford and the Marquis of Tavistock, [[Social Victorians/People/de Soveral|M. de Soveral, the Portuguese Minister]], and Viscount and Viscountess Falmouth.

Others to be picked out in the crowd were Consuelo Duchess of Manchester, Viscountess Coke and Mrs. Ellis, Lady Archibald Campbell and her son, Mrs. Grenander, Lord and Lady Lilford, Mr. and Mrs. Edward Stonor, Mrs. [[Social Victorians/People/Bourke|Algernon Bourke]], Mr. Algernon Grosvenor, and Mr. and Mrs. Hwfa Williams.

The royal party took a great interest in the contests, and especially applauded the Swedish couple in their graceful evolutions. Their Majesties remained over an hour, the royal party taking their departure shortly after five.&lt;ref&gt;&quot;Social &amp; Personal.&quot; ''Daily Express'' 14 February 1902, Friday: 4 [of 8], Cols. 3c–4b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19020214/088/0004. Print p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

===March 1902===
The last time Bret Harte and Arthur Collins saw each other: &quot;They dined at the Royal Thames Yacht Club, and Collins found his 'poor old friend' 'saldly aged and broken, but genial and kind as ever.' They sat an hour at a music hall and Harte wrote afterwards to thank Collins for having 'forced him out.'&quot; (Nissen, Axel. Bret Harte: Prince and Pauper. Jackson, MS: U P of Mississippi, 2000: 262)
===April 1902===
====9 April 1902, Wednesday====
According to a letter to Lady Gregory, [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] dictated &quot;2000 words in an hour and a half&quot; &quot;to a typewriter; he was working on his novel (Wade 370). At this point, a typewriter was a person who used the machine called typewriter to type.
====10 April 1902, Thursday====
[[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote to Lady Gregory from 18 Woburn Buildings about working on his novel &quot;-- dictating to a typewriter&quot; (Wade 370).

===May 1902===
====5 May 1902, Monday====
Bret Harte died. Arthur Collins does not seem to have been there at his death; “his dear friend Madame Van de Velde and her attendants” were, though (Pemberton, T. Edgar. The Life of Bret Harte. Dodd, Meade, 1903. http://books.google.com/books?id=eZMOAAAAMAAJ). Not sure when the funeral occurred, but he is buried “in quiet Frimly churchyard,” (341) and &lt;quote&gt;In accordance with his well-known views on such subjects the funeral was a very simple one. Among the few who followed him to his ivy-lined grave were Mrs. Bret Harte, his son and daughter-in-law, Mr. and Mrs. Francis King Harte, his daughter, Miss Ethel Harte, Madame Van de Velde, Colonel Collins, Mr. A.S. Boyd, and a small cluster of grief-stricken friends.&lt;/quote&gt; (Pemberton, T. Edgar. The Life of Bret Harte. Dodd, Meade, 1903. http://books.google.com/books?id=eZMOAAAAMAAJ (accessed November 2014).

====8 May 1902, Thursday====
&lt;quote&gt;On Thursday, May 8, 1902, in the squat, mid-Victorian church of St. Peter's in the Surrey village of Frimley, a group of about twenty people had come to show their final respects to Francis Bret Harte. Outside it was raining steadily . In the subdued light from the stained-glass windows, one cold discern a small group at the front of the church consisting of Anna Harte, her son Frank, her daughter-in-law Aline, and her daughter Ethel. Another small group was formed around Madame Van de Velde, including one of her unmarried daughters, Miss Norris (the sister of her son-in-law Richard Norris), and Mrs. Clavering Lyne. Of Harte's closest friend, only Arthur Collins and Alexander Stuart Boyd were present. Pemberton had written to Frank the day before that he wished to attend the funeral but that in his &quot;deplorable state of health&quot; it was impossible for him to travel. Beside the small group of family and old friends, the rest of the people who heard the service conducted by the rector of Frimley, Reverend W. Basset, were recent acquaintances from among the local gentry. As one newspaper noted: &quot;The funeral was of the simplest possible character and the phrase 'this our brother' had a peculiar poignancy, for, though a group of villagers stood in the rain under the trees as the hearse arrived, there were few in the church, who had not the right to call Mr. Bret Harte friend.&quot; The simplicity of the service was in keeping with Bret Harte's wishes.&lt;/quote&gt; (Nissen, Axel. Bret Harte: Prince and Pauper. Jackson, MS: U P of Mississippi, 2000: 263)
===June 1902===
Summer 1902: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until Yeats bought the Tower at Ballylee. (I got this from Wade?)
====3 June 1902, Tuesday====
[[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote Arnold Dolmetsch, asking him to &quot;chair ... a lecture he [was] soon to give&quot;: &quot;You are the only one, I suppose, in the world now, who knows anything about the old music that was half speech, and I need hardly say that neither [[Social Victorians/People/Florence Farr|Miss Farr]] nor myself, could have done anything in this matter of speaking to notes without your help&quot; (Campbell 142).
====7-9 June 1902, Saturday-Monday====
The [[Social Victorians/People/Warwick|Earl and Countess of Warwick]] hosted a house party: &lt;quote&gt;The Earl and Countess of Warwick entertained a distinguished house party from Saturday to yesterday, including the Grand Duke Michael of Russia and the Countess of Torby, the Earl and Countess of Craven, the Earl and Countess of Kilmorey, Earl Cairns, Lord and Lady Savile, Lord Chesham, Sir Frederick and Lady Milner, Colonel and Lady Gwendoline Colvin. Lady Margaret Orr-Ewing, Lady Eva Dugdale. Mrs. Kenneth Wilson, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Right Hon. H. Chaplin, M.P., Hon. H. Stonor, Mr. J. Pease, M.P., Captain Brinton, and Captain J. Forbes.&lt;/quote&gt; (1902-06-10 Manchester Courier and Lancashire General Advertiser)

====10 June 1902, Tuesday====
[[Social Victorians/People/Florence Farr|Florence Farr]]'s first public performance in which she &quot;recit[ed] to her own accompaniment on the psaltery was at the Hall of Clifford's Inn, Fleet Street, on 10 June 1902 (Campbell 144, n. 18).

==== 12 June 1902, Thursday====
12 June 1902: &lt;quote&gt;Thursday, the 12th inst., being the grand day of Trinity term at Gray's-inn, the Treasurer (Mr. Herbert Reed, K.C.) and the Masters of the Bench entertained at dinner the following guests: The Right Hon. Lord Strathoona and Mount Royal, the Right Hon. Lord Avebury, the Right Hon. H. H. Asquith, K.C, M.P., the Right Hon. Sir Frank Lascelles, G.C.B. (British Minister at Berlin), General Sir Edward Brabant, K.C.B., the Right Hon. Sir Edward Carson (Solicitor-General), Sir Squire Bancroft, Colonel Alfred Egerton, C.B. (Equerry to H.R.H. the Duke of Connaught), Mr. Austen Chamberlain,M.P., Colonel Royds, M.P., and Mr. Frank Dicksee, R.A. The Benchers present in addition to the Treasurer were H.R H. the Duke of Connaught, Lord Ashbourne, Lord Shand, Mr. Henry Griffith, Sir Arthur Collins, K.C, Mr. Hugh Shield, K.C, His Honour Judge Bowen Rowlands, K.C, Mr. James Sheil, Mr. Arthur Beetham, Mr. John Rose, Mr. Paterson, Mr. Mulligan, K.C, Mr. Mattinson, K.C, Mr. Macaskie, K.C., Mr. C. A. Russell, K.C., Mr. Montague Lush, K.C., Mr. Dicey, C B., Mr. Barnard, Mr. H. C. Richards, K.C., M.P., Mr. Duke, K.C., M.P., Sir Julian Salomons, K.C., with the Preacher (the Rev. Canon C. J. Thompson, D.D.).&lt;/quote&gt; (The Solicitor's Journal and Reporter. June 21, 1902. Volume XLVI. 1901-1902 [November 2, 1901, to October 25, 1902]: 588. Google Books: http://books.google.com/books?id=9T84AQAAIAAJ&amp;pg=PA588).

====26 June 1902, Thursday====
Edward VII crowned King of England. 26 June 1902.

There was apparently a regular celebration of Arthur Collins' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in May 1902, so the celebration with Harte present did not take place this year. Did it take place at all?

===July 1902===
====3 July 1902, Thursday====
[[Social Victorians/People/Mathers|MacGregor and Moina Mathers]] were living at 28 Rue Saint Vincent, Buttes Montmartre, Paris (Howe 244).

===September 1902===
''Tristan and Isolde'' at the Covent Garden.
====25 September 1902, Thursday====
&quot;There were no winter performances of opera at Covent Garden in those times .... In 1902 an autumnal series was added, and there were several Wagner nights, the last of which was on Thursday, 25 September, when Philip Brozel and Blanch Marchesi were starred in ''Tristan and Isolda'' with Marie Alexander as Brangane&quot; (Baring-Gould II 704, n. 14, quoting Rolfe Boswell).

===October 1902===

==== 24 October 1902, Friday ====
The ''Daily Express'' reported on the annual opening of the Prince's ice-skating rink, revealing who had an interest in skating:&lt;blockquote&gt;PRINCE’S RINK OPENS.

The first ice of the season was skated upon yesterday. It was the carefully-prepared ice which Mr. H. W. Page and Mr. Nightingale offer to the members of Prince’s Skating Club, in Knightsbridge, and was in grand condition.

The [[Social Victorians/People/Bourke|Hon. Algernon Bourke]] opened the rink for the seventh season, and in the afternoon and evening the West End patronized the popular club to skate or to lounge to the pleasant strains of the Viennese band.

[[Social Victorians/People/Princess Louise|Princess Louise]] is again at the head of the ladies’ committee, with the [[Social Victorians/People/Portland|Duchess of Portland]] and [[Social Victorians/People/Londonderry|Marchioness of Londonderry]] as co-members, and Lord Edward Cecil and many other well-known skaters are identified with the committee work.

The skating hours are from 9.30 to 1 and 3 to 7, and on Sundays 3 to 7 only.&lt;ref&gt;&quot;Prince's Rink Opens.&quot; ''Daily Express'' 25 October 1902, Saturday: 5 [of 8], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004848/19021025/132/0005.&lt;/ref&gt;&lt;/blockquote&gt;

====31 October 1902, Friday====
Halloween.
===November 1902===
====5 November 1902, Wednesday====
Guy Fawkes Day.

==== 8 November 1902, Saturday ====
The Earl and Countess of Warwick hosted a shooting party at Easton Lodge:&lt;blockquote&gt;The [[Social Victorians/People/Warwick|Earl and Countess of Warwick]] are entertaining a large party at Easton Lodge this week-end for [?] shooting, and among their guests are the Grand Duke Michael of Russia and Countess Torby, the Duc d'Alba, the Duke of Sutherland, Earl Howe, Earl Cairns, Lord Dalmeny, Lord Herbert Vane-Tempest, the Hon. John and Lady [Choely?] Scott-Montagu, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Right Hon. Henry Chaplin, M.P., General and Mrs. Arthur Paget, and Miss Leila Paget, Miss Naylor, Miss Deacon, and Mr W. M. Low.&lt;ref&gt;&quot;Guests at Easton Lodge.&quot; ''Birmingham Mail'' 08 November 1902, Saturday: 2 [of 6], Col. 8b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000644/19021108/091/0002. Print title: ''Birmingham Daily Mail''; p. 2.&lt;/ref&gt;&lt;/blockquote&gt;

====29 November 1902, Saturday====
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]]’s cousin, Lady Hartopp, was involved in a divorce case:
&lt;blockquote&gt;Society Women in a Law Court Case.

Mr. Justice Barnes’s Court is now crowded by society people. What is the strange fascination which brings elegantly dressed ladies, accustomed to luxurious surroundings and all the external refinements of life, to sit for hours in stuffy court, where the accommodation is all the plainest, and the surroundings are none too attractive. It would need some assurance to invite a Belgravian Countess, or the wife of Mayfair Millionaire to spend the morning under such conditions unless there were the attraction of a very strong piece of scandal. One could not presume to suggest she should attend Missionary meeting, or social reform movement, under any such conditions. At least I must confess that I never heard of one being packed with a West End crowd as the Court just now. Of course it cannot be mere idle curiosity. Our higher education for girls must have cured Mother Eve’s failing long ago. Cynics suggest that it is the survival in our highly-civilised modern conditions of that instinct of the wild creature which incites attack on the wounded or injured fellow. Wild birds will sometimes peck injured bird to death. Are these fair and soft-voiced ladies animated by the same spirit when they throng witness the ordeal through which a woman of their own class is passing?

The Latest Divorce Case.

Lady Hartopp, the heroine of the story which has been occupying the tongues and thoughts of the upper ten thousand for the last 48 hours, is a member of a well-known and wealthy family, and is herself remarkable for her beauty. Her two sisters are as famous for their charms as herself, and society has given them many flattering titles. The daughters of Mr. C. H. Wilson, the great shipowner, whose sails are on every sea, are as favoured by Fortune as Venus. Miss Muriel Wilson, the society beauty, is a cousin of Lady Hartopp, and Lady Chesterfield is her sister. It was at Tranby Croft, near Hull, the residence of Mr. and Mrs. Arthur Wilson, that the famous baccarat case occurred some years ago. Lady Hartopp is the niece of Mr. Arthur Wilson, and no doubt recollects that incident, and all the consequent stir. It attracted all the more notice at the time, because the then Prince of Wales had taken part in the game; but the Prince, who had nothing to be ashamed of, with characteristic straightforwardness, asked to go into the box and state all he knew. (1902-11-29 Norwich Mercury)&lt;/blockquote&gt;

===December 1902===

==== 9 December 1902, Tuesday ====
&quot;Severe weather&quot; did not prevent Lady Eva Wyndham's &quot;at home&quot; from being a success:&lt;blockquote&gt;Lady Wyndham-Quin's &quot;At Home.&quot;

The severe weather proved to be no detriment to the many visitors who had accepted Lady Eva Wyndham-Quin's invitation to an &quot;at home&quot; at the Welch Industrial depot on Tuesday afternoon, and the admirers and purchasers of the fascinating Christmas gifts were numerous. Lady Eva received her quests wearing a coat of Persian paw and a white feather toque, whilst her two tittle daughters the Misses Olein and Kethlean Wyndham-Quin wore pelisses and hats of pale blue Welsh frieze, trimmed with grebe. Amongst those present were Lady George Hamilton, all in black; Lady Brassey, wearing a lovely sable cape; the [[Social Victorians/People/Bourke|Hon. Mrs Algernon Bourke]], in a fur coat and a black picture hat; and the Hon. Mrs Herbert, of Llanever; Mrs Brynmor Jones was fall of her coming visit to Paris to see her young daughter, and Mrs Richard Helme came with her son, Mr Ernest Helme. Mrs Brenton and her sister, Mrs Ashurst Morris, were also present, as were Lady Eafield, the Dowager Lady Hylton, Lady Dennison Pender [Ponder?], and Lady Blanche Conyngham. Mrs Grinnell Milne brought Miss Murray end Mrs Shelley Bontens, and Mrs James Head came in for a few minutes. Everybody bought largely and the Welsh Christmas cards were an attractive feature, as were some artistic muff chains. Another specimen of Welsh lace sent by Miss Jenkins, of Denbighshire, was much admired and resembles Irish lace both in style and design.&lt;ref&gt;&quot;A Lady Correspondent.&quot; &quot;Society in London.&quot; ''South Wales Daily News'' 11 December 1902, Thursday: 4 [of 8], Col. 5a [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000919/19021211/082/0004. Print p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

====16 December 1902, Tuesday====
A poem satirizing Florence Farr and Arnold Dolmetsch was published in ''Punch''.
====25 December 1902, Thursday====
Christmas Day
====26 December 1902, Friday====
Boxing Day
===Works Cited===
*[1902-06-10 Manchester Courier and Lancashire General Advertiser] &quot;Court and Personal.&quot; Manchester Courier and Lancashire General Advertiser 10 June 1902, Tuesday: 5 [of 10], Col. 3c [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000206/19020610/033/0006 (accessed July 2019).
*[1902-11-29 Norwich Mercury] &quot;Society Women in a Law Court Case.&quot; And &quot;The Latest Divorce Case.&quot; Norwich Mercury 29 November 1902, Saturday: 5 [of 12], Col. 1b [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001669/19021129/072/0005 (accessed July 2019).

==1903==
From sometime in 1891 to sometime in 1903 Eduoard de Reszke was &quot;a leading bass&quot; at the New York Metropolitan Opera (Baring-Gould II 112, n. 114).

&quot;[I]n England in 1903, gramophone distinctly meant the Berliner-Gramophon &amp; Typewriter disc machine, while cyclinder machines were known as phonographs or graphophones &quot; (Baring-Gould II 745, n. 15).

Gerald Balfour was &quot;largely responsible for getting the important Land Acts of 1903 under way&quot; (O'Connor 163).
===January 1903===
====1 January 1903, Thursday, New Year's Day====
====3 January 1903, Saturday====
Madame Troncey was doing a portrait of [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] (Wade 392).
===June 1903===
Summer 1903: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

=== August–September 1903 ===

==== 20 and 25 August and 3 September 1903 ====
The 1903 America's Cup yacht race in New York Harbor with Nathaniel Herreshoff's ''Reliance'' for the US and Sir Thomas Lipton's ''Shamrock III'' for the UK,&lt;ref&gt;{{Cite journal|date=2022-09-11|title=1903 America's Cup|url=https://en.wikipedia.org/w/index.php?title=1903_America%27s_Cup&amp;oldid=1109663279|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/1903_America%27s_Cup.&lt;/ref&gt; the 12th challenge for the cup and &quot;the most expensive Cup challenge in history.&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite web|url=https://www.americascup.com/history/26_LIPTONS-THIRD-CHALLENGE|title=LIPTON’S THIRD CHALLENGE|last=Cup|first=America's|website=37th America's Cup|language=en|access-date=2024-07-02}} https://www.americascup.com/history/26_LIPTONS-THIRD-CHALLENGE.&lt;/ref&gt; The first race was run on 20 August 1903, the 2nd on 25 August and the 3rd on 3 September.&lt;ref name=&quot;:0&quot; /&gt; Because the ''Reliance'' won the first 3 races, the best 3-out-of-5 race ended after the 3rd one.

===October 1903===
Sometime in October 1903, [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s &quot;The Adventure of the Empty House,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 331).
====31 October 1903, Saturday====
Halloween.
===November 1903===
Sometime in November 1903 Arthur Conan Doyle's &quot;The Adventure of the Norwood Builder,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 415).
====5 November 1903, Thursday====
Guy Fawkes Day
===December 1903===
Sometime in December 1903 Arthur Conan Doyle's &quot;The Adventure of the Dancing Men,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 529).
====16 December 1903, Wednesday====
&quot;On 16 December, Punch satirised an activity in which Dolmetsch was concerned. [[Social Victorians/People/Florence Farr|Florence Farr]] was acting as secretary for a newly-formed fellowship known as 'The Dancers', a body whose aim was to 'fight the high and powerful devil, solemnity'. In a poem entitled L'Allegro up to date, the final stanza is devoted to Dolmetsch:
&lt;poem&gt;:The old forgotten dancing-lore,
:The steps we cannot understand,
:DOLMETSCH agrees to take in hand,
:These on the well-trod stage anon,
:When next our learned sock is on,
:We’ll show, while ARNOLD, Fancy’s child,
:Tootles his native wood-wind wild.&lt;/poem&gt;
This verse is curiously prophetic for Dolmetsch had not yet introduced the recorder into his concerts, although he occasionally included a flute. Dolmetsch did know something of the steps of the old dances but it was his wife who later researched the subject most thoroughly and wrote two books on the subject.&quot; (Campbell 151–52)

===25 December 1903, Friday===
Christmas Day
====26 December 1903, Saturday====
Boxing Day
===Works Cited===
*Baring-Gould.
*Campbell.

==1904==
===January 1904===
Sometime in January 1904 [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]]'s &quot;The Adventure of the Solitary Cyclist,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 399).
===March 1904===
Sometime in March 1904 Arthur Conan Doyle's &quot;The Adventure of Black Peter,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 384).

===April 1904===
Sometime in April 1904, Arthur Conan Doyle's &quot;The Adventure of Charles Augustus Milverton,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 558, n. 1, and 559).

===June 1904===
Sometime in June 1904 Arthur Conan Doyle's &quot;The Adventure of the Three Students,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 370).

Summer 1904: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

===July 1904===
Sometime in July 1904, Arthur Conan Doyle's &quot;The Adventure of the Golden Pince-Nez,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 351).
===August 1904===
Sometime in August 1904, Arthur Conan Doyle's &quot;The Adventure of the Missing Three-Quarter,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 476).

===September 1904===
Sometime in September 1904, Arthur Conan Doyle's &quot;The Adventure of the Abbey Grange,&quot; illustrated by Sidney Paget, was published in the ''Strand'' (Baring-Gould II 491).

==1905==
===April 1905===
====3 April 1905, Monday====
[[Social Victorians/People/William Butler Yeats|W. B. Yeats]] wrote to Lady Gregory from Dublin, saying he had &quot;dictated a rough draft of a new Grania second act to Moore's typewriter&quot; (Wade 368).
===June 1905===
Summer 1905: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).
===July 1905===
====10 July 1905, Monday====
1905 July 10, the Austro-Hungarian Ambassador hosted a dinner party:&lt;blockquote&gt;The Austro-Hungarian Ambassador entertained the Duke and Duchess of Connaught and Princess Patricia of Connaught at dinner at the Embassy in Belgrave-square on Monday evening. There were also present the Spanish Ambassador and Mme. Bernabé, the United States Ambassador and Mrs. and Miss Whitelaw Reid, Princess Hohenlohe, Prince Francis of Teck, Princess Teano, the Earl of Essex, the Earl and Countess of Crewe, Viscount Villiers, Viscount Errington, Viscount Newry, Mrs. J. Leslie, [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]], Mr. R. Graham, Mrs. Astor, Lady Maud Warrender, Prince Furstenburg, Count Szenchenyi, Captain A. Meade, and Miss Pelly and Colonel Murray in attendance on the Duke and Duchess.&lt;ref&gt;&quot;Court Circular.&quot; ''Times'', 12 July 1905, p. 7. ''The Times Digital Archive'', http://tinyurl.galegroup.com/tinyurl/AHRNq6. Accessed 20 June 2019.&lt;/ref&gt;&lt;/blockquote&gt;

==== Last week of July, 1905 ====
Lady Cadogan hosted a children's party at Chelsea House:&lt;blockquote&gt;Lady Cadogan’s children’s party last week at Chelsea House was one of the prettiest sights imaginable. Her grandchildren, the little Chelseas, came to help entertain the guests, and nearly all the smart women in London brought their small folk. One of loveliest little girls present was Daphne Bourke, Mrs. [[Social Victorians/People/Bourke|Algernon Bourke]]’s only child; and Lady De Trafford’s young daughter Violet was much admired, and Lady Maud Ramsden’s little people were among daintiest of the small children.&lt;ref&gt;&quot;Court and Social News.&quot; ''Belfast News-Letter'' 01 August 1905, Tuesday: 7 [of 10], Col. 6b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/19050801/157/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

===October 1905===

==== 1905 October 14, Saturday ====
A &quot;send-off dinner&quot; for Jerome K. Jerome before his trip to the U.S. occurred at the Garrick Club &quot;the other evening&quot; before October 14:&lt;blockquote&gt;Jerome K. Jerome has undertaken a six months lecturing tour in the United States. I believe that this tour will be a great success, particularly when the Americans come to realise that Mr. Jerome is not only a humorous writer but a brilliant, serious writer with very genuine pathos. His appeal on this side has not, perhaps, gone home to the English people as much as it should, but the quick-witted Americans will not be slow to recognise his talents of both kinds, nor will they fail to appreciate the significance of the fact that the other evening a send-off dinner was given to Mr. Jerome at the Garrick Club. The hosts of the evening were Mr. Pett Ridge and Mr. W. W. Jacobs, which shows that there is no such thing as literary jealousy among our best humorists.

The presence of quite a galaxy of novelists to the dinner to Mr. Jerome, including Mr. Barrie, Sir Arthur Conan Doyle, Mr. Max Pemberton, Mr. H. G. Wells, Mr. G. B. Burgin, Mr. Arthur Morrison, and Mr. Israel Zangwill, serve to indicate the existence of a pleasant brotherhood among the writers of fiction. The readers of ''Three Men in a Boat'' may be interested to know that there were also present Mr. Jerome's companions in that famous journey — Mr. Carl Hentschel and Mr. C. Wingrove.

When I have named further the presence of three artists in Mr. A. S. Boyd, Mr. John Hassall, and Mr. Will Owen, and two journalists in Dr. Robertson Nicoll and [[Social Victorians/People/Rook|Mr. Clarence Rook]], I have given some record of an exceedingly pleasant dinner party. The essential point, however, of this enumeration of names is that many of them are among the most highly honoured of Englishmen in the United States, and that thus Mr. Jerome cannot fail to reap additional benefit from this dinner so thoughtfully given in his honour by Mr. Jacobs and Mr. Pett Ridge.&lt;ref&gt;S., C. K. &quot;A Literary Letter.&quot; ''The Sphere'' 14 October 1905, Saturday: 16 [of 20], Col. 2a–c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001861/19051014/022/0016. Print p. 56.&lt;/ref&gt;&lt;/blockquote&gt;

===November 1905===
Sometime in November 1905, &quot;Arnold Dolmetsch was again asked to provide music for a Been Greet season in New York — an engagement that brought about his first meeting with two young actors on their first American tour, Sybil Thorndike, and her brother, Russell&quot; (Campbell 169). Dolmetsch's return to the US; was [[Social Victorians/People/Horniman|Annie Horniman]] still with the Thorndikes?

==1906==
===March 1906===
====5 March 1906====
&quot;Mr. Frederick John Horniman, who died on March 5, in his seventy-first year, was the son of that well-known Quaker and tea-merchant, John Horniman, who made a magnificent fortune by retailing tea in air-tight packets, and, like his father, devoted both time and wealth to charitable objects. A great traveller, both for business and pleasure, Mr. Horniman gathered togther an admirable collection of curios, and this is housed at Forest Hill in the museum that bears his name. His private benefactions were also large. Mr. Horniman, who was a Liberal, sat in two Parliaments, representing Penrhyn and Falmouth Boroughs in one. He did not seek re-election in January last.&quot; (&quot;The World's News.&quot; Illustrated London News (London, England), Saturday, March 10, 1906; pg. 338; Issue 3490, Col. C)
===June 1906===
Summer 1906: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

==1907==
===April 1907===
April 1907, [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] went to Italy with Lady Gregory (Harper 80 28).
===June 1907===
Summer 1907: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

'''1907 June 22, Saturday'''
The annual dinner of the Correctors of the Press was held at De Keyser's Royal Hotel:&lt;blockquote&gt;The London Association of Correctors of the Press held their annual dinner at De Keyser’s Royal Hotel on Saturday. The Chairman was the Lord Mayor, and among his supporters were Sir John Cockburn, Colonel David Bruce, Colonel Earl Church, Lieutenant-Colonel Alsager Pollock, Sheriff Dunn, Mr. J. W. Cleland, M.P., Mr. R. Donald, Mr. T. Seccombe, Mr. Francis H. Skrine, Major H. F. Trippel, Mr. Walter Haddon, Mr. W. Pett Ridge, Mr. W. H. Helm, Mr. R. Warwick Bond, Mr. F. W. Rudler, Major Vane Stow, [[Social Victorians/People/Rook|Mr. Clarence Rook]], Mr. J. Randall (Chairman of the Association), Mr. Foxen, and Mr. Feldwick. Proposing the toast of &quot;Literature,” Mr. W. H. Helm speculated as to what would follow the banning of &quot;Mary Barton&quot; by the Education Committee of the London County Council. In his opinion &quot;The Swiss Family Robinson&quot; was a more immoral book, because beyond any other work it had fostered the Micawber view of life. (Laughter.) The LORD MAYOR [init caps large, rest sm, throughout], submitting the toast of &quot;The Readers' Pension Fund,” apologised for appearing in morning dress. The reason was that he had been to the King’s Garden Party at Windsor, and whlle he was returning to London by motor something burst. (Laughter.) Only that morning he had arrived from Berlin, where he learned some lessons useful to people who give dinners. When the Oberburgomeister of Berlin proposed the health of, say, the Lord Mayor of London, there was an end of the business. He did not push forward the Houses of Parliament, the Navy and Army, or even Literature. (Laughter.) Being a practical people the Germans when they met for a particular purpose applied themselves to no other, and the English would well to copy them. (Hear, hear.) Mr. J. RANDALL said that last year the Association helped five readers and one reader’s widow to pensions, and this year it had done the same for two readers and two widows. One of the men assisted last March had taught himself Greek, Arabic, and Sanscrit, and in leisure moments amused himself by making object glasses for microscopes and telescopes. At this very gathering there was a printer’s reader who was Hebrew scholar. (Hear, hear.) With regard to finance Mr. Randall was happy to say that this dinner would enable the Association to establish a fourth pension. (Cheers.) The Lord Mayor, [[Social Victorians/People/Borthwick|Lord Glenesk]] (President of the Readers' Pensions Committee), the Clothworkers’ Company, and the Cutlers’ Company had contributed ten guineas each, and the total addition to the fund resulting from the dinner was £l90. During the evening excellent entertainment was provided by Miss Helena Foxen, Miss Kathleen Dwyer, Mr. T. C. Bell, Mr. P. E. Syrett, Mr. Prank Rhodes, and Mr. E. Croft-Williams, the last-named being the hon. musical director.&lt;ref&gt;&quot;Correctors of the Press.&quot; ''Morning Post'' 24 June 1907, Monday: 4 [of 14], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19070624/074/0004. Print p. 4.&lt;/ref&gt;&lt;/blockquote&gt;

===November 1907===
====10 November 1907====
&lt;quote&gt;On 10 November, Dolmetsch, 'awfully tired and disquieted with overwork', writes to Horne, 'longing for Florence'.

7, Bayley Street&lt;br /&gt;W.C.&lt;br /&gt;My concert went very well last night. Melodie quite distinguished herself, and a sister of [[Social Victorians/People/George Bernard Shaw|Bernard Shaw]] Lucy Carr Shaw sang delightfully. …&lt;br /&gt;But Symmons [sic] … did not go before 1 o'cl. and yet, by the first post this morning, I got a charming poem on Rameau. … He must have spent all night on it.&lt;/quote&gt; (Campbell 120)

==1908==
In 1908 Sidney Paget died in 1908 in some &quot;untimely&quot; fashion (Baring-Gould II 239).

=== April 1908 ===

==== 1908 April 9, Thursday ====
The Provisional Committee for the Shakespeare Memorial demonstration at the Lyceum Theatre met at the Hôtel Métropole:&lt;blockquote&gt;SHAKESPEARE MEMORIAL.

A meeting of the Provisional Committee for the forthcoming Shakespeare Memorial demonstration at the Lyceum Theatre was held yesterday at the Hôtel Métropole. Mr. T. P. O’Connor, M.P., presided, and there were present : The Earl of Lytton, Mr. Percy Alden, M.P., Mr. Henry Ainley, Mr. Percy Ames, Mr. Robert Barr, Mr. Arthur à Beckett, Mr. Austin Brereton, Mr. Acton Bond (General Director of the British Empire Shakespeare Society), Mr. Dion Boucicault, Mrs. Bateman-Crowe, Professor Boss, Mr. Norreys Connell, Mr. W. M. Crook, Mr. John Cutler, K.C., Mr. J. Comyns Carr, Mr. Ernest Carpenter, the Rev. P. H. Ditchfleld, Mr. Robert Donald, Mr. A. C. Forster Boulton, M.P., Mr. and Mrs. Laurence Gomme, Mr. A. A. Gardiner, Mr. C. T. Hunt (hon. secretary London Shakespeare League), Mr. Laurence Housman, Mr. J. A. Hobson. Mr. Ford Madox Hueffer, Mr. Selwyn Image, Mr. Henry Arthur Jones, Mr. Jerome K. Jerome, Mr. Frederick Kerr, Miss Gertrude Kingston, Professor Knight, Mr. Matheson Lang, the Hon. Mrs. Alfred Lyttelton, Miss Lillah McCarthy, Mr. Justin Huntly McCarthy, Colonel Henry Mapleson, Dr. Gilbert Murray, Mr. T. Fairman Ordish, Mr. A. W. Pinero, Mr. Ernest Rhys, [[Social Victorians/People/Rook|Mr. Clarence Rook]], the Rev. J. Cartmel Robinson, Mr. George Radford, M.P., Mr. Clement Shorter, Mr. Otto Salimann (hon. secretary of the Elizabethan Society), [[Social Victorians/People/George Bernard Shaw|Mr. Bernard Shaw]], Mr. H. W. Smith, Mr. Herbert Trench, [[Social Victorians/People/Todhunter|Dr. Todhunter]], and Mr. James Welch. It was agreed that the Lyceum demonstration should take place in May, and a resolution should be moved in favour of the establishment of a National Theatre as a memorial to Shakespeare.&lt;ref&gt;&quot;Shakespearea Memorial.&quot; ''Morning Post'' 10 April 1908, Friday: 7 [of 12], Col. 3c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/19080410/126/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

===June 1908===
Summer 1908: [[Social Victorians/People/William Butler Yeats|W. B. Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

==== 1908 June 21, Sunday ====
Very large demonstration for women's suffrage in Hyde Park coming from &quot;seven points in London.&quot;&lt;blockquote&gt;WOMAN'S VOTE.
SUFFRAGISTS' GREAT MARCH TO HYDE PARK TODAY.

WHITE DEMONSTRATION.

AMUSING ADDRESS TO M.P.'s. FROM RIVER LAUNCH.

From seven points in London to-day seven big prossesions will march to Park, and there jointly demand the Parliamentary franchise for women.

The whole town will be alive with demonstrating suffragists. The streets will resound with the cry of &quot;Votes for Women.&quot; In Hyde Park eighty speakers will voice the demand from twenty platforms.

London has been divided into districts for the purposes of the mighty demonstration, and each of theee has an assembling place, from which the processions will move off to Hyde Park, as given in the following official list: —

A. — Euston-road. — Form up at 12 o'clock, east of Euston Station. Start at 1 p.m. March via Euston-road, Portland-place, Upper Regent-street, Oxford-street, to the Marble Arch.

B. — Trafalgar-square. — Form up 12.30. Start 1.30. March via Pall Mall, Regent-street, Piccadilly, Berkeley-street, and Mount-street to the Grosvenor Gate.

C. — Victoria Embankment. [sic] Form up 12.30. Start from Westminster Bridge 1.30. March via Victorla-street, Grosvenor-place, to Hyde Park Corner.

D. — Chelsea Embankment. — Form up 12.30. Start 1.30. March via Oakley-street, King's-road, Sloane-square, Sloane-street to Albert Gate.

E. — Kensington High-street. — Form up 1 o'clock. Start 1.30. March via Kensington into the Alexandra Gate of the Park.

F. — Paddington Station. — Form up 1 p.m. Start 2 p.m. March via Victoria Gate into Hyde Park.

G. — Marylebone-road. — Form up 12.30. Start 1.30. March via Seymour-place, Seymour-street, and Into the Park close to the Marble Arch.

The demonstrators will come from all parts of the country, some seventy special trains being run from the big towns in the provinces. These will be met at the London stations by white-garbed &quot;Captains&quot; and &quot;Stewards,&quot; and their occupants marshalled in proper divisions.

Literature and the drama will be represented in several of the processions. Mr. and Mrs. Bernard Shaw will join in Trafalgar-square, and so will Mr. Pett Ridge. Starting from Euston-road will be a coach carrying Mrs. Parkhurst, Miss Beatrice Harraden, Mrs. Mona Caird, and Miss Elizabeth Robins.

Mrs. Israel Zangwill will chaperon a party on a coach from the Thames Embankment, which will include Professor and Mrs. Ayrton, Madame Sarah Grand, Miss Lillah McCarthy (Mrs. Granville Barker), Miss Marian McCarthy, Mr. Lucien Wolf, Professor Perry, F.R.S. (scientist), Mrs. H. G. Wells, Mrs. Alice Meynell, and Suffragist leaders from Sweden, Finland, and Norway. In Finland women not only have the vote, but they sit in Parliament. Madame Stromberg, from that country, is now in London attending the Horse Show at Olympia, and will be present at to-day's demonstration. Mr. H. Nevinson and Mr. H. N. Brailsford will walk in the Embankment procession.

On the Kensington four-in-hand coach will be:— [[Social Victorians/People/Rook|Mrs. Clarence Rook]], Mrs. Jopling Rowe, Mlle. Stavance (Norwegian editor and authoress), Mrs. French Sheldon, F.R.G.S., and Miss Christine Silver. 

...

In addition to seven four-horse coaches — one for each procession — there will be more than sixty brakes, filled with country suffragists, and elaborately decorated. [Story continues.]&lt;ref&gt;&quot;Women's Vote. Suffragists' Great March to Hyde Park To-day. White Demonstration. Amusing Address to M.P.'s from River Launch.&quot; ''Lloyd's Weekly Newspaper'' 21 June 1908, Sunday: 1 [of 28], Col. 1a–c [of 5], 2, Col. 5. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003216/19080621/002/0001. Print p. 1.&lt;/ref&gt;&lt;/blockquote&gt;

===Works Cited===
*Baring-Gould

==1909==
===January 1909===
====1 January 1909====
Rev. [[Social Victorians/People/Ayton|W. A. Ayton]] died (Howe 85 10-11).
===June 1909===
Summer 1909: W. B. Yeats summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).

== Bibliography ==
#&quot;Calendar for the Year 1900.&quot; Jumk.de Webprojects. https://kalender-365.de/public-holidays.php?yy=1900. Accessed November 2023.
#Howe

== Footnotes ==
&lt;references /&gt;</text>
      <sha1>ox1bajze6hxobepeu1woekuds07dio7</sha1>
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  </page>
  <page>
    <title>Workings of gcc and ld in plain view</title>
    <ns>0</ns>
    <id>285384</id>
    <revision>
      <id>2692108</id>
      <parentid>2692003</parentid>
      <timestamp>2024-12-15T23:46:07Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>/* Linking Libraries */</comment>
      <origin>2692108</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3346" sha1="2xjzlv3v9nyuzpx2m64ufngnr4t163t" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241214.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])

* Shared Libraries ([[Media:LIB.2A.Shared.20241216.pdf |pdf]])

==== Managing Libraries ====

* Shared Library Names ([[Media:MNG.1A.Names.20241214.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
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    <title>User:Dc.samizdat/Rotations</title>
    <ns>2</ns>
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    <revision>
      <id>2692094</id>
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      <timestamp>2024-12-15T22:52:38Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Spinors */ cite A.P. Goucher 2019 on Spin Groups</comment>
      <origin>2692094</origin>
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      <text bytes="102112" sha1="aerwj4o6gqa1pajghm4reu8ayujo4g9" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2023 - November 2024}}

&lt;blockquote&gt;'''Abstract:''' The physical universe is properly visualized as a Euclidean space of four or more orthogonal spatial dimensions. Atoms are 4-polytopes, and stars are 4-balls of atomic plasma. A galaxy is a hollow 3-sphere with these objects distributed on its surface; the black hole at its center is the 4-ball of empty space it surrounds. The observable universe of galaxies may be visualized as a 4-sphere expanding radially from an origin point at velocity &lt;math&gt;c&lt;/math&gt;, the propagation speed of light through 3-space, which is also the invariant velocity of all mass-carrying objects through 4-space. The propagation speed of light through 4-space &lt;math&gt;c_4&lt;/math&gt; is actually &lt;math&gt;c &lt; c_4 &lt; 2c&lt;/math&gt;. This view is compatible with the theories of special and general relativity, and with the quantum mechanical atomic theory. It explains those theories as expressions of intrinsic symmetries.&lt;/blockquote&gt;

== Symmetries ==
It is common to speak of nature as a web, and so it is, the great web of our physical experiences. Every web must have its root systems somewhere, and nature in this sense must be rooted in the symmetries which underlie physics and geometry, the [[W:Group (mathematics)|mathematics of groups]].{{Sfn|Conway|Burgiel|Goodman-Strauss|2008}}

As I understand [[W:Noether's theorem|Noether's theorem]] (which is not mathematically), hers is the deepest meta-theory of nature yet, deeper than [[W:Theory of relativity|Einstein's relativity]] or [[W:Evolution|Darwin's evolution]] or [[W:Euclidean geometry|Euclid's geometry]]. It finds that all fundamental findings in physics are based on conservation laws which can be laid at the doors of distinct [[W:symmetry group |symmetry group]]s.{{Efn|[[W:Coxeter group|Coxeter theory]] is for geometry what Noether's theorem is for physics. [[W:Coxeter|Coxeter]] showed that Euclidean geometry is based on conservation laws that obey the principle of relativity and correspond to distinct symmetry groups.}} Thus all fundamental systems in physics, as examples [[W:quantum chromodynamics|quantum chromodynamics]] (QCD) the theory of the strong force binding the atomic nucleus and [[W:quantum electrodynamics|quantum electrodynamics]] (QED) the theory of the electromagnetic force, each have a corresponding symmetry [[W:group theory|group theory]] of which they are an expression. As I understand [[W:Coxeter group|Coxeter group]] theory (which is not mathematically), the symmetry groups underlying physics seem to have an expression in a [[W:Euclidean space|Euclidean space]] of four [[W:dimension|dimension]]s, that is, they are [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]]. Therefore as I understand that geometry (which is entirely by synthetic rather than algebraic methods), the [[W:Atom|atom]] seems to have a distinct Euclidean geometry, such that atoms and their constituent particles are four-dimensional objects, and nature can be understood in terms of their [[W:group action|group actions]], including centrally [[W:rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]].

== The geometry of the atomic nucleus ==

In [[W:Euclidean 4-space|Euclidean four dimensional space]], an [[W:atomic nucleus|atomic nucleus]] is a [[24-cell]], the regular 4-polytope with [[W:Coxeter group#Symmetry groups of regular polytopes|𝔽&lt;sub&gt;4&lt;/sub&gt; symmetry]]. Nuclear shells are concentric [[W:3-sphere|3-sphere]]s occupied (fully or partially) by the orbits of this 24-point [[#The 6 regular convex 4-polytopes|regular convex 4-polytope]]. An actual atomic nucleus is a rotating four dimensional object. It is not a ''rigid'' rotating 24-cell, it is a kinematic one, because the nucleus of an actual atom of any [[W:nucleon number|nucleon number]] contains a distinct number of orbiting vertices which may be in different isoclinic rotational orbits. These moving vertices never describe a static 24-cell at any single instant in time, though their orbits do all the time. The physical configuration of the nucleus as a 24-cell can be reduced to the [[W:kinematics|kinematics]] of the orbits of its constituents. The geometry of the atomic nucleus is therefore strictly [[W:Euclidean geometry#19th century|Euclidean]] in four dimensional space.

=== Rotations ===
The [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]] of the convex [[W:regular 4-polytope|regular 4-polytope]]s are usually described as discrete rotations of a rigid object. For example, the rigid [[24-cell]] can rotate in a [[24-cell#Hexagons|hexagonal]] (6-vertex) central [[24-cell#Planes of rotation|plane of rotation]]. A 4-dimensional [[24-cell#Isoclinic rotations|''isoclinic'' rotation]] (as distinct from a [[24-cell#Simple rotations|''simple'' rotation]] like the ones that occur in 3-dimensional space) is a ''diagonal'' rotation in multiple [[W:Clifford parallel|Clifford parallel]] [[24-cell#Geodesics|central planes]] of rotation at once. It is diagonal because it is a [[W:SO(4)#Double rotations|double rotation]]: in addition to rotating in parallel (like wheels), the multiple planes of rotation also tilt sideways (like coins flipping) into each other's central planes. Consequently, the path taken by each vertex is a [[24-cell#Helical hexagrams and their isoclines|twisted helical circle]], rather than the ordinary flat circle a vertex follows in a simple rotation. In a rigid 4-polytope rotating isoclinically, ''all'' the vertices lie in one or another of the parallel planes of rotation, so all of them move in parallel along Clifford parallel twisting circular paths. [[24-cell#Clifford parallel polytopes|Clifford parallel planes]] are not parallel in the normal sense of parallel planes in three dimensions; the vertices are all moving in different directions around the [[W:3-sphere|3-sphere]]. In one complete 360° isoclinic revolution, a rigid 4-polytope turns itself inside out.

This is sufficiently different from the simple rotations of rigid bodies in our 3-dimensional experience that a precise [[24-cell|detailed description]] enabling the reader to visualize it runs to many pages and illustrations, with many accompanying pages of explanatory notes on basic phenomena that arise only in 4-dimensional space: [[24-cell#Squares|completely orthogonal planes]], [[24-cell#Hexagons|Clifford parallelism]] and [[W:Hopf fibration|Hopf fiber bundles]], [[24-cell#Helical hexagrams and their isoclines|isoclinic geodesic paths]], and [[24-cell#Double rotations|chiral (mirror image) pairs of rotations]], among other complexities. Moreover, the characteristic rotations of the various regular 4-polytopes are all different; each is a surprise. [[#The 6 regular convex 4-polytopes|The 6 regular convex 4-polytopes]] have different numbers of vertices (5, 8, 16, 24, 120, and 600 respectively) and those with fewer vertices occur inscribed in those with more vertices (generally), with the result that the more complex 4-polytopes subsume the kinds of rotations characteristic of their less complex predecessors, as well as each having a characteristic kind of rotation not found in their predecessors. [[W:Euclidean geometry#Higher dimensions|Four dimensional Euclidean space]] is more complicated (and more interesting) than three dimensional space because there is more room in it, in which unprecedented things can happen. It is much harder for us to visualize, because the only way we can experience it is in our imaginations; we have no body of ''sensory'' experience in 4-dimensional space to draw upon.

For that reason, descriptions of isoclinic rotations usually begin and end with rigid rotations: [[24-cell#Isoclinic rotations|for example]], all 24 vertices of a rigid 24-cell rotating in unison, with 6 vertices evenly spaced around each of 4 Clifford parallel twisted circles.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} But that is only the simplest case. [[W:Kinematics|Kinematic]] 24-cells (with moving parts) are even more interesting (and more complicated) than the rigid 24-cell. 

To begin with, when we examine the individual parts of the rigid 24-cell that are moving in an isoclinic rotation, such as the orbits of individual vertices, we can imagine a case where fewer than 24 point-objects are orbiting on those twisted circular paths at once. [[24-cell#Reflections|For example]], if we imagine just 8 point-objects, evenly spaced around the 24-cell at [[24-cell#Reciprocal constructions from 8-cell and 16-cell|the 8 vertices that lie on the 4 coordinate axes]], and rotate them isoclinically along exactly the same orbits they would take in the above-mentioned rotation of a rigid 24-cell, in the course of a single 360° rotation the 8 point-objects will trace out the whole 24-cell, with just one point-object reaching each of the 24 vertices just once, and no point-object colliding with any other at any time.

That is still an example of a rigid object in a single distinct isoclinic rotation: a rigid 8-vertex object (called the 4-[[W:orthoplex|orthoplex]] or [[16-cell]]) performing the characteristic rotation of the 24-cell. But we can also imagine ''combining'' distinct rotations. What happens when multiple point-objects are orbiting at once, but do ''not'' all follow the Clifford parallel paths characteristic of the ''same'' distinct rotation? What happens when we combine orbits from distinct rotations characteristic of different 4-polytopes, for example when different rigid 4-polytopes are concentric and rotating simultaneously in their characteristic ways? What kinds of such hybrid rotations are possible without collisions? What sort of [[Kinematics of the cuboctahedron|kinematic polytopes]] do they trace out, and how do their [[24-cell#Clifford parallel polytopes|component parts]] relate to each other as they move? Is there (sometimes) some kind of mutual stability amid their lack of combined rigidity? Visualizing isoclinic rotations (rigid and otherwise) allows us to explore questions of this kind of [[W:kinematics|kinematics]], and where dynamic stabilites arise, of [[W:kinetics|kinetics]].

=== Isospin ===
A [[W:Nucleon|nucleon]] is a [[W:proton|proton]] or a [[W:neutron|neutron]]. The proton carries a positive net [[W:Electric charge|charge]], and the neutron carries a zero net charge. The proton's [[W:Mass|mass]] is only about 0.13% less than the neutron's, and since they are observed to be identical in other respects, they can be viewed as two states of the same nucleon, together forming an isospin doublet ({{nowrap|''I'' {{=}} {{sfrac|1|2}}}}). In isospin space, neutrons can be transformed into protons and conversely by actions of the [[W:SU(2)|SU(2)]] symmetry group. In nature, protons are very stable (the most stable particle known); a proton and a neutron are a stable nuclide; but free neutrons decay into protons in about 10 or 15 seconds.

According to the [[W:Noether theorem|Noether theorem]], [[W:Isospin|isospin]] is conserved with respect to the [[W:strong interaction|strong interaction]].&lt;ref name=Griffiths2008&gt;{{cite book |author=Griffiths, David J. |title=Introduction to Elementary Particles |edition=2nd revised |publisher=WILEY-VCH |year=2008 |isbn=978-3-527-40601-2}}&lt;/ref&gt;{{rp|129–130}} Nucleons are acted upon equally by the strong interaction, which is invariant under rotation in isospin space.

Isospin was introduced as a concept in 1932 by [[W:Werner Heisenberg|Werner Heisenberg]],&lt;ref&gt;
{{cite journal
 |last=Heisenberg |first=W. |author-link=W:Werner Heisenberg
 |year=1932
 |title=Über den Bau der Atomkerne
 |journal=[[W:Zeitschrift für Physik|Zeitschrift für Physik]]
 |volume=77 |issue=1–2 |pages=1–11
 |doi=10.1007/BF01342433
 |bibcode = 1932ZPhy...77....1H
 |s2cid=186218053
 |language=de}}&lt;/ref&gt; well before the 1960s development of the [[W:quark model|quark model]], to explain the symmetry of the proton and the then newly discovered neutron. Heisenberg introduced the concept of another conserved quantity that would cause the proton to turn into a neutron and vice versa. In 1937, [[W:Eugene Wigner|Eugene Wigner]] introduced the term &quot;isospin&quot; to indicate how the new quantity is similar to spin in behavior, but otherwise unrelated.&lt;ref&gt;
{{cite journal
 |last=Wigner |first=E. |author-link=W:Eugene Wigner
 |year=1937
 |title=On the Consequences of the Symmetry of the Nuclear Hamiltonian on the Spectroscopy of Nuclei
 |journal=[[W:Physical Review|Physical Review]]
 |volume=51
 |pages=106–119
 |doi=10.1103/PhysRev.51.106
 |bibcode = 1937PhRv...51..106W
 |issue=2
}}&lt;/ref&gt; Similar to a spin-1/2 particle, which has two states, protons and neutrons were said to be of isospin 1/2. The proton and neutron were then associated with different isospin projections ''I''&lt;sub&gt;3&lt;/sub&gt;&amp;nbsp;=&amp;nbsp;+1/2 and −1/2 respectively.

Isospin is a different kind of rotation entirely than the ordinary spin which objects undergo when they rotate in three-dimensional space. Isospin does not correspond to a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] in any space (of any number of dimensions). However, it does seem to correspond exactly to an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in a Euclidean space of four dimensions. Isospin space resembles the [[W:3-sphere|3-sphere]], the [[W:Elliptical space#Elliptic space (the 3D case)|curved 3-dimensional space]] that is the surface of a [[W:4-ball (mathematics)#In Euclidean space|4-dimensional ball]].

=== Spinors ===
[[File:Spinor on the circle.png|thumb|upright=1.5|A spinor visualized as a vector pointing along the [[W:Möbius band|Möbius band]], exhibiting a sign inversion when the circle (the &quot;physical system&quot;) is continuously rotated through a full turn of 360°.]][[W:Spinors|Spinors]] are [[W:representation of a Lie group|representations]] of a [[W:spin group|spin group]], which are [[W:Double covering group|double cover]]s of the [[W:special orthogonal group|special orthogonal groups]]. The spin group Spin(4) is the double cover of [[W:SO(4)|SO(4)]], the group of rotations in 4-dimensional Euclidean space. [[600-cell#Fibrations of isocline polygrams|Isoclines]], the helical geodesic paths followed by points under isoclinic rotation, correspond to spinors representing Spin(4).

Spinors can be viewed as the &quot;square roots&quot; of [[W:Section (fiber bundle)|cross sections]] of [[W:vector bundle|vector bundle]]s; in this correspondence, a fiber bundle of isoclines (of a distinct isoclinic rotation) is a cross section (inverse bundle) of a fibration of great circles (in the invariant planes of that rotation).

A spinor can be visualized as a moving vector on a Möbius strip which transforms to its negative when continuously rotated through 360°, just as [[24-cell#Helical hexagrams and their isoclines|an isocline can be visualized as a Möbius strip]] winding twice around the 3-sphere, during which [[24-cell#Isoclinic rotations|720° isoclinic rotation]] the rigid 4-polytope turns itself inside-out twice.{{Sfn|Goucher|2019|loc=Spin Groups}} Under isoclinic rotation, a rigid 4-polytope is an isospin-1/2 object with two states.

=== Isoclinic rotations in the nucleus ===
Isospin is regarded as a symmetry of the strong interaction under the [[W:Group action (mathematics)|action]] of the [[W:Lie group|Lie group]] [[W:SU(2)|SU(2)]], the two [[W:eigenstate|states]] being the [[W:Up quark|up flavour]] and [[W:Down quark|down flavour]]. A 360° isoclinic rotation of a rigid [[W:nuclide|nuclide]] would transform its protons into neutrons and vice versa, exchanging the up and down flavours of their constituent [[W:quarks|quarks]], by turning the nuclide and all its parts inside-out (or perhaps we should say upside-down). Because we never observe this, we know that the nucleus is not a ''rigid'' polytope undergoing isoclinic rotation.

If the nucleus ''were'' a rigid object, nuclides that were isospin-rotated 360° would be isoclinic mirror images of each other, isospin +1/2 and isospin −1/2 states of the whole nucleus. We don't see whole nuclides rotating as a rigid object, but considering what would happen if they ''were'' rigid tells us something about the geometry we must expect inside the nucleons. One way that an isospin-rotated neutron could become a proton would be if the up quark and down quark were a left and right mirror-image pair of the same object; exchanging them in place would turn each down-down-up neutron into an up-up-down proton. But the case cannot be quite that simple, because the up quark and the down quark are not mirror-images of the same object: they have very different mass and other incongruities.

Another way an isospin-rotated neutron could be a proton would be if the up and down quarks were asymmetrical kinematic polytopes (not indirectly congruent mirror-images, and not rigid polytopes), rotating within the nucleus in different ''hybrid'' orbits. By that we mean that they may have vertices orbiting in rotations characteristic of more than one 4-polytope, so they may change shape as they rotate. In that case their composites (protons and neutrons) could have a symmetry not manifest in their components, but emerging from their combination.

....

=== Hybrid isoclinic rotations ===
The 24-cell has [[24-cell#Isoclinic rotations|its own characteristic isoclinic rotations]] in 4 Clifford parallel hexagonal planes (each intersecting 6 vertices), and also inherits the [[16-cell#Rotations|characteristic isoclinic rotations of its 3 Clifford parallel constituent 16-cells]] in 6 Clifford parallel square planes (each intersecting 4 vertices). The twisted circular paths followed by vertices in these two different kinds of rotation have entirely different geometries. Vertices rotating in hexagonal invariant planes follow [[24-cell#Helical hexagrams and their isoclines|helical geodesic curves whose chords form hexagrams]], and vertices rotating in square invariant planes follow [[24-cell#Helical octagrams and their isoclines|helical geodesic curves whose chords form octagrams]].

In a rigid isoclinic rotation, ''all'' the [[24-cell#Geodesics|great circle polygons]] move, in any kind of rotation. What distinguishes the hexagonal and square isoclinic rotations is the invariant planes of rotation the vertices stay in. The rotation described [[#Rotations|above]] (of 8 vertices rotating in 4 Clifford parallel hexagonal planes) is a single hexagonal isoclinic rotation, not a kinematic or hybrid rotation.

A ''kinematic'' isoclinic rotation in the 24-cell is any subset of the 24 vertices rotating through the same angle in the same time, but independently with respect to the choice of a Clifford parallel set of invariant planes of rotation and the chirality (left or right) of the rotation. A ''hybrid'' isoclinic rotation combines moving vertices from different kinds of isoclinic rotations, characteristic of different regular 4-polytopes. For example, if at least one vertex rotates in a square plane and at least one vertex rotates in a hexagonal plane, the kinematic rotation is a hybrid rotation, combining rotations characteristic of the 16-cell and characteristic of the 24-cell.

As an example of the simplest hybrid isoclinic rotation, consider a 24-cell vertex rotating in a square plane, and a second vertex, initially one 24-cell edge-length distant, rotating in a hexagonal plane. Rotating isoclinically at the same rate, the two moving vertices will never collide where their paths intersect, so this is a ''valid'' hybrid rotation.

To understand hybrid rotations in the 24-cell more generally, visualize the relationship between great squares and great hexagons. The [[24-cell#Squares|18 great squares]] occur as three sets of 6 orthogonal great squares,{{Efn|name=six orthogonal planes of the Cartesian basis}} each [[16-cell#Coordinates|forming a 16-cell]]. The three 16-cells are completely disjoint{{Efn|name=completely disjoint}} and [[24-cell#Clifford parallel polytopes|Clifford parallel]]: each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}).{{Efn|name=three isoclinic 16-cells}} The 18 square great circles are crossed by 16 hexagonal great circles; each [[24-cell#Hexagons|hexagon]] has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two [[24-cell#Triangles|great triangles]] inscribed in each great hexagon (occupying its alternate vertices, with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking three completely disjoint great squares, one from each of the three completely disjoint 16-cells''.{{Efn|There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}} Isoclinic rotations take the elements of the 4-polytope to congruent [[24-cell#Clifford parallel polytopes|Clifford parallel elements]] elsewhere in the 4-polytope. The square rotations do this ''locally'', confined within each 16-cell: for example, they take great squares to other great squares within the same 16-cell. The hexagonal rotations act ''globally'' within the entire 24-cell: for example, they take great squares to other great squares in ''different'' 16-cells. The [[16-cell#Helical construction|chords of the square rotations]] bind the 16-cells together internally, and the [[24-cell#Helical hexagrams and their isoclines|chords of the hexagonal rotations]] bind the three 16-cells together.

....

=== Color ===
When the existence of quarks was suspected in 1964, [[W:Oscar W. Greenberg|Greenberg]] introduced the notion of color charge to explain how quarks could coexist inside some [[W:hadron|hadron]]s in [[W:quark model#The discovery of color|otherwise identical quantum states]] without violating the [[W:Pauli exclusion principle|Pauli exclusion principle]]. The modern concept of [[W:color charge|color charge]] completely commuting with all other charges and providing the strong force charge was articulated in 1973, by [[W:William A. Bardeen|William Bardeen]], [[W:de:Harald Fritzsch|Harald Fritzsch]], and [[W:Murray Gell-Mann|Murray Gell-Mann]].&lt;ref&gt;{{cite conference |author1=Bardeen, W. |author2=Fritzsch, H. |author3=Gell-Mann, M. |year=1973 |title=Light cone current algebra, ''π''&lt;sup&gt;0&lt;/sup&gt; decay, and ''e''&lt;sup&gt;+&lt;/sup&gt; ''e''&lt;sup&gt;&amp;minus;&lt;/sup&gt; annihilation |arxiv=hep-ph/0211388 |editor=Gatto, R. |book-title=Scale and conformal symmetry in hadron physics |page=[https://archive.org/details/scaleconformalsy0000unse/page/139 139] |publisher=[[W:John Wiley &amp; Sons|John Wiley &amp; Sons]] |isbn=0-471-29292-3 |bibcode=2002hep.ph...11388B |url-access=registration |url=https://archive.org/details/scaleconformalsy0000unse/page/139 }}&lt;/ref&gt;&lt;ref&gt;{{cite journal |title=Advantages of the color octet gluon picture |journal=[[W:Physics Letters B|Physics Letters B]] |volume=47 |issue=4 |page=365 |year=1973 |last1=Fritzsch |first1=H. |last2=Gell-Mann |first2=M. |last3=Leutwyler |first3=H. |doi=10.1016/0370-2693(73)90625-4 |bibcode=1973PhLB...47..365F |citeseerx=10.1.1.453.4712}}&lt;/ref&gt;

Color charge is not [[W:electric charge|electric charge]]; the whole point of it is that it is a quantum of something different. But it is related to electric charge, through the way in which the three different-colored quarks combine to contribute fractional quantities of electric charge to a nucleon. As we shall see, color is not really a separate kind of charge at all, but a partitioning of the electric charge into [[24-cell#Clifford parallel polytopes|Clifford parallel subspaces]].

The [[W:Color charge#Red, green, and blue|three different colors]] of quark charge might correspond to three different 16-cells, such as the three disjoint 16-cells inscribed in the 24-cell. Each color might be a disjoint domain in isospin space (the space of points on the 3-sphere).{{Efn|The 8 vertices of each disjoint 16-cell constitute an independent [[16-cell#Coordinates|orthonormal basis for a coordinate reference frame]].}} Alternatively, the three colors might correspond to three different fibrations of the same isospin space: three different ''sequences'' of the same total set of discrete points on the 3-sphere. These alternative possibilities constrain possible representations of the nuclides themselves, for example if we try to represent nuclides as particular rotating 4-polytopes. If the neutron is a (8-point) 16-cell, either of the two color possibilities might somehow make sense as far as the neutron is concerned. But if the proton is a (5-point) 5-cell, only the latter color possibility makes sense, because fibrations (which correspond to distinct isoclinic left-and-right rigid rotations) are the ''only'' thing the 5-cell has three of. Both the 5-cell and the 16-cell have three discrete rotational fibrations. Moreover, in the case of a rigid, isoclinically rotating 4-polytope, those three fibrations always come one-of-a-kind and two-of-a-kind, in at least two different ways. First, one fibration is the set of invariant planes currently being rotated through, and the other two are not. Second, when one considers the three fibrations of each of these 4-polytopes, in each fibration two isoclines carry the left and right rotations respectively, and the third isocline acts simply as a Petrie polygon, the difference between the fibrations being the role assigned to each isocline.

If we associate each quark with one or more isoclinic rotations in which the moving vertices belong to different 16-cells of the 24-cell, and the sign (plus or minus) of the electric charge with the chirality (right or left) of isoclinic rotations generally, we can configure nucleons of three quarks, two performing rotations of one chirality and one performing rotations of the other chirality. The configuration will be a valid kinematic rotation because the completely disjoint 16-cells can rotate independently; their vertices would never collide even if the 16-cells were performing different rigid square isoclinic rotations (all 8 vertices rotating in unison). But we need not associate a quark with a [[16-cell#Rotations|rigidly rotating 16-cell]], or with a single distinct square rotation.

Minimally, we must associate each quark with at least one moving vertex in each of three different 16-cells, following the twisted geodesic isocline of an isoclinic rotation. In the up quark, that could be the isocline of a right rotation; and in the down quark, the isocline of a left rotation. The chirality accounts for the sign of the electric charge (we have said conventionally as +right, −left), but we must also account for the quantity of charge: +{{sfrac|2|3}} in an up quark, and −{{sfrac|1|3}} in a down quark. One way to do that would be to give the three distinct quarks moving vertices of {{sfrac|1|3}} charge in different 16-cells, but provide up quarks with twice as many vertices moving on +right isoclines as down quarks have vertices moving on −left isoclines (assuming the correct chiral pairing is up+right, down−left).

Minimally, an up quark requires two moving vertices (of the up+right chirality).{{Efn|Two moving vertices in one quark could belong to the same 16-cell. A 16-cell may have two vertices moving in the same isoclinic square (octagram) orbit, such as an antipodal pair (a rotating dipole), or two vertices moving in different square orbits of the same up+right chirality.{{Efn|There is only one [[16-cell#Helical construction|octagram orbit]] of each chirality in each fibration of the 16-cell, so two octagram orbits of the same chirality cannot be Clifford parallel (part of the same distinct rotation). Two vertices right-moving on different octagram isoclines in the same 16-cell is a combination of two distinct rotations, whose isoclines will intersect: a kinematic rotation. It can be a valid kinematic rotation if the moving vertices will never pass through a point of intersection at the same time. Octagram isoclines pass through all 8 vertices of the 16-cell, and all eight isoclines (the left and right isoclines of four different fibrations) intersect at ''every'' vertex.}} However, the theory of [[W:Color confinement|color confinement]] may not require that two moving vertices in one quark belong to the same 16-cell; like the moving vertices of different quarks, they could be drawn from the disjoint vertex sets of two different 16-cells.}} Minimally, a down quark requires one moving vertex (of the down−left chirality). In these minimal quark configurations, a proton would have 5 moving vertices and a neutron would have 4.

....

=== Nucleons ===
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|[[W:Branko Grünbaum|Grünbaum's]] rotationally symmetrical 5-set Venn diagram, 1975. It is the [[5-cell]]. Think of it as an [[W:Nuclear magnetic resonance|NMR image]] of the 4-dimensional proton in projection to the plane.]]
The proton is a very stable mass particle. Is there a stable orbit of 5 moving vertices in 4-dimensional Euclidean space? There are few known solutions to the 5-body problem, and fewer still to the [[W:n-body problem|{{mvar|n}}-body problem]], but one is known: the ''central configuration'' of {{mvar|n}} bodies in a space of dimension {{mvar|n}}-1. A [[W:Central configuration|central configuration]] is a system of [[W:Point particle|point masses]] with the property that each mass is pulled by the combined attractive force of the system directly towards the [[W:Center of mass|center of mass]], with acceleration proportional to its distance from the center. Placing three masses in an equilateral triangle, four at the vertices of a regular [[W:Tetrahedron|tetrahedron]], five at the vertices of a regular [[5-cell]], or more generally {{mvar|n}} masses at the vertices of a regular [[W:Simplex|simplex]] produces a central configuration [[W:Central configuration#Examples|even when the masses are not equal]]. In an isoclinic rotation, all the moving vertices orbit at the same radius and the same speed. Therefore if any 5 bodies are orbiting as an isoclinically rotating regular 5-cell (a rigid 4-simplex figure undergoing isoclinic rotation), they maintain a central configuration, describing 5 mutually stable orbits.

Unlike the proton, the neutron is not always a stable particle; a free neutron will decay into a proton. A deficiency of the minimal configurations is that there is no way for this [[W:beta minus decay|beta minus decay]] to occur. The minimal neutron of 4 moving vertices described [[#Color|above]] cannot possibly decay into a proton by losing moving vertices, because it does not possess the four up+right moving vertices required in a proton. This deficiency could be remedied by giving the neutron configuration 8 moving vertices instead of 4: four down−left and four up+right moving vertices. Then by losing 3 down−left moving vertices the neutron could decay into the 5 vertex up-down-up proton configuration.{{Efn|Although protons are very stable, during [[W:stellar nucleosynthesis|stellar nucleosynthesis]] two H&lt;sub&gt;1&lt;/sub&gt; protons are fused into an H&lt;sub&gt;2&lt;/sub&gt; nucleus consisting of a proton and a neutron. This [[W:beta plus decay|beta plus &quot;decay&quot;]] of a proton into a neutron is actually the result of a rare high-energy collision between the two protons, in which a neutron is constructed. With respect to our nucleon configurations of moving vertices, it has to be explained as the conversion of two 5-point 5-cells into a 5-point 5-cell and an 8-point 16-cell, emitting two decay products of at least 1-point each. Thus it must involve the creation of moving vertices, by the conversion of kinetic energy to point-masses.}} A neutron configuration of 8 moving vertices could occur as the 8-point 16-cell, the second-smallest regular 4-polytope after the 5-point 5-cell (the hypothesized proton configuration).

It is possible to double the neutron configuration in this way, without destroying the charge balance that defines the nucleons, by giving down quarks three moving vertices instead of just one: two −left vertices and one +right vertex. The net charge on the down quark remains −{{sfrac|1|3}}, but the down quark becomes heavier (at least in vertex count) than the up quark, as in fact its mass is measured to be.

A nucleon's quark configuration is only a partial specification of its properties. There is much more to a nucleon than what is contained within its three quarks, which contribute only about 1% of the nucleon's energy. The additional 99% of the nucleon mass is said to be associated with the force that binds the three quarks together, rather than being intrinsic to the individual quarks separately. In the case of the proton, 5 moving vertices in the stable orbits of a central configuration (in one of the [[5-cell#Geodesics and rotations|isoclinic rotations characteristic of the regular 5-cell]]) might be sufficient to account for the stability of the proton, but not to account for most of the proton's energy. It is not the point-masses of the moving vertices themselves which constitute most of the mass of the nucleon; if mass is a consequence of geometry, we must look to the larger geometric elements of these polytopes as their major mass contributors. The quark configurations are thus incomplete specifications of the geometry of the nucleons, predictive of only some of the nucleon's properties, such as charge.{{Efn|Notice that by giving the down quark three moving vertices, we seem to have changed the quark model's prediction of the proton's number of moving vertices from 5 to 7, which would be incompatible with our theory that the proton configuration is a rotating regular 5-cell in a central configuration of 5 stable orbits. Fortunately, the actual quark model has nothing at all to say about moving vertices, so we may choose to regard that number as one of the geometric properties the quark model does not specify.}} In particular, they do not account for the forces binding the nucleon together. Moreover, if the rotating regular 5-cell is the proton configuration and the rotating regular 16-cell is the neutron configuration, then a nucleus is a complex of rotating 5-cells and 16-cells, and we must look to the geometric relationship between those two very different regular 4-polytopes for an understanding of the nuclear force binding them together.

The most direct [[120-cell#Relationships among interior polytopes|geometric relationship among stationary regular 4-polytopes]] is the way they occupy a common 3-sphere together. Multiple 16-cells of equal radius can be compounded to form each of the larger regular 4-polytopes, the 8-cell, 24-cell, 600-cell, and 120-cell, but it is noteworthy that multiple regular 5-cells of equal radius cannot be compounded to form any of the other 4-polytopes except the largest, the 120-cell. The 120-cell is the unique intersection of the regular 5-cell and 16-cell: it is a compound of 120 regular 5-cells, and also a compound of 75 16-cells. All regular 4-polytopes except the 5-cell are compounds of 16-cells, but none of them except the largest, the 120-cell, contains any regular 5-cells. So in any compound of equal-radius 16-cells which also contains a regular 5-cell, whether that compound forms some single larger regular 4-polytope or does not, no two of the regular 5-cell's five vertices ever lie in the same 16-cell. So the geometric relationship between the regular 5-cell (our proton candidate) and the regular 16-cell (our neutron candidate) is quite a distant one: they are much more exclusive of each other's elements than they are distantly related, despite their complementary three-quark configurations and other similarities as nucleons. The relationship between a regular 5-cell and a regular 16-cell of equal radius is manifest only in the 120-cell, the most complex regular 4-polytope, which [[120-cell#Geometry|uniquely embodies all the containment relationships]] among all the regular 4-polytopes and their elements.

If the nucleus is a complex of 5-cells (protons) and 16-cells (neutrons) rotating isoclinically around a common center, then its overall motion is a hybrid isoclinic rotation, because the 5-cell and the 16-cell have different characteristic isoclinic rotations, and they have no isoclinic rotation in common.{{Efn|The regular 5-cell does not occur inscribed in any other regular 4-polytope except one, the 600-vertex 120-cell. No two of the 5 vertices of a regular 5-cell can be vertices of the same 16-cell, 8-cell, 24-cell, or 600-cell. The isoclinic rotations characteristic of the regular 5-cell maintain the separation of its 5 moving vertices in 5 disjoint Clifford-parallel subspaces at all times. The [[16-cell#Rotations|isoclinic rotation characteristic of the 16-cell]] maintains the separation of its 8 moving vertices in 2 disjoint Clifford-parallel subspaces (completely orthogonal great square planes) at all times. Therefore, in any hybrid rotation of a concentric 5-cell and 16-cell, at most one 5-cell subspace (containing 1 vertex) might be synchronized with one 16-cell subspace (containing 4 vertices), such that the 1 + 4 vertices they jointly contain occupy the same moving subspace continually, forming a rigid 5-vertex polytope undergoing some kind of rotation. If in fact it existed, this 5-vertex rotating rigid polytope would not be [[5-cell#Geometry|not a 5-cell, since 4 of its vertices are coplanar]]; it is not a 4-polytope but merely a polyhedron, a [[W:square pyramid|square pyramid]].}}

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=== Nuclides ===

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=== Quantum phenomena ===

The Bell-Kochen-Specker (BKS) theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a space of three or more dimensions can be given by exhibiting a finite set of lines through the origin that cannot each be colored black or white in such a way that (i) no two orthogonal lines are both black, and (ii) not all members of a set of ''d'' mutually orthogonal lines are white.{{Efn|&quot;The Bell-Kochen-Specker theorem rules out the existence of deterministic noncontextual hidden variables theories. A proof of the theorem in a Hilbert space of dimension d ≥ 3 can be given by exhibiting a finite set of rays [9] that cannot each be assigned the value 0 or 1 in such a way that (i) no two orthogonal rays are both assigned the value 1, and (ii) not all members of a set of d mutually orthogonal rays are assigned the value 0.&quot;{{Sfn|Waegell|Aravind|2009|loc=2. The Bell-Kochen-Specker (BKS) theorem}}|name=BKS theorem}}

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=== Motion ===
What does it mean to say that an object moves through space? Coxeter group theory provides precise answers to questions of this kind. A rigid object (polytope) moves by distinct transformations, changing itself in each discrete step into a congruent object in a different orientation and position.

....

== Galilean relativity in a space of four orthogonal dimensions ==

Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions.

General relativity is just Galilean relativity in a general space of four orthogonal dimensions, e.g. Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;, spherical 4-space &lt;math&gt;S^4&lt;/math&gt;, or any orthogonal 4-manifold.

Light is just reflection. Gravity (and all force) is just rotation. Both motions are just group actions, expressions of intrinsic symmetries. That is all of physics.

Every observer properly sees himself as stationary and the universe as a sphere with himself at the center. The curvature of these spheres is a function of the rate at which causality evolves, and it can be measured by the observer as the speed of light.

=== Special relativity is just Galilean relativity in a Euclidean space of four orthogonal dimensions ===
Perspective effects occur because each observer's ordinary 3-dimensional space is only a curved manifold embedded in 4-dimensional Euclidean space, and its curvature complicates the calculations for him (e.g., he sometimes requires Lorentz transformations). But if  all four spatial dimensions are considered, no Lorentz transformations are required (or permitted) except when you want to calculate a projection, or a shadow, that is, how things will appear from a three-dimensional viewpoint (not how they really are).{{Sfn|Yamashita|2023}} The universe really has four spatial dimensions, and space and time behave just as they do in classical 3-vector space, only bigger by one dimension. It is not necessary to combine 4-space with time in a spacetime to explain 4-dimensional perspective effects at high velocities, because 4-space is already spatially 4-dimensional, and those perspective effects fall out of the 4-dimensional Pythagorean theorem naturally, just as perspective does in three dimensions. The universe is only strange in the ways the Euclidean fourth dimension is strange; but that does hold many surprises for us. Euclidean 4-space is much more interesting than Euclidean 3-space, analogous to the way that 3-space is much more interesting than 2-space. But all Euclidean spaces are dimensionally analogous. Dimensional analogy itself, like everything else in nature, is an exact expression of intrinsic symmetries.

=== General relativity is just Galilean relativity in a general space of four orthogonal dimensions ===
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=== Physics ===
....

=== Thoreau's spherical relativity ===
Every observer may properly see himself as stationary and the universe as a 4-sphere with himself at the center observing it, perceptually equidistant from all points on its surface, including his own ''physical'' location which is one of those surface points, distinguished to him but not the center of anything. This statement of the principle of relativity is compatible with Galileo's relativity of uniformly moving objects in ordinary space, Einstein's special relativity of inertial reference frames in 4-dimensional spacetime, Einstein's general relativity of all reference frames in curved, non-Euclidean spacetime, and Coxeter's relativity of orthogonal group actions in Euclidean spaces of any number of dimensions.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q&lt;sup&gt;2&lt;/sup&gt; in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q&lt;sup&gt;2&lt;/sup&gt;. By the same principle, we can view any QT or Q&lt;sup&gt;2&lt;/sup&gt; as an isoclinic (equi-angled) Q&lt;sup&gt;2&lt;/sup&gt; by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations, which intuitively we might see follows from the fact that any transformation from one inertial reference frame to another is expressable as a [[W:SO(4)|rotation in 4-dimensional Euclidean space]].|name=Cayley's rotation factorization into two isoclinic reference frame transformations}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} It should be known as Thoreau's spherical relativity, since the first precise written statement of it appears in 1849: &quot;The universe is a sphere whose center is wherever there is intelligence.&quot;{{Sfn|Thoreau|1849|p=349|ps=; &quot;The universe is a sphere whose center is wherever there is intelligence.&quot; [Contemporaneous and independent of [[W:Ludwig Schlafli|Ludwig Schlafli]]'s pioneering work enumerating the complete set of regular polytopes in any number of dimensions.{{Sfn|Coxeter|1973|loc=§7. Ordinary Polytopes in Higher Space; §7.x. Historical remarks|pp=141-144|ps=; &quot;Practically all the ideas in this chapter ... are due to Schläfli, who discovered them before 1853 — a time when Cayley, Grassman and Möbius were the only other people who had ever conceived the possibility of geometry in more than three dimensions.&quot;}}]}}

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== Conclusions==

=== Spherical relativity ===

We began our inquiry by wondering why physical space should be limited to just three dimensions (why ''three''). By visualizing the universe as a Euclidian space of four dimensions, we recognize that relativistic and quantum phenomena are natural consequences of symmetry group operations (including reflections and rotations) in four orthogonal dimensions. We should not then be surprised to see that the universe does not have just four dimensions, either. Physical space must bear as many dimensions as we need to ascribe to it, though the distinct phenomena for which we find a need to do so, in order to explain them, seem to be fewer and fewer as we consider higher and higher dimensions. To laws of physics generally, such as the principle of relativity in particular, we should always append the phrase &quot;in Euclidean spaces of any number of dimensions&quot;. Laws of physics should operate in any flat Euclidean space &lt;math&gt;R^n&lt;/math&gt; and in its corresponding spherical space &lt;math&gt;S^n&lt;/math&gt;.

The first and simplest sense in which we are forced to contemplate a fifth dimension is to accommodate our normal idea of time. Just as Einstein was forced to admit time as a dimension, in his four-dimensional spacetime of three spatial dimensions plus time, for some purposes we require a fifth time dimension to accompany our four spatial dimensions, when our purpose is orthogonal to (in the sense of independent of) the four spatial dimensions. For example, if we theorize that we observe a finite homogeneous universe, and that it is a Euclidean 4-space overall, we may prefer not to have to identify any distinct place within that 4-space as the center where the universe began in a big bang. To avoid having to pick a distinct place as the center of the universe, our model of it must be expanded, at least to be a ''spherical'' 4-dimensional space with the fifth radial dimension as time. Essentially, we require the fifth dimension in order to make our homogeneous 4-space finite, by wrapping it around into a 4-sphere. But perhaps we can still resist admitting the fifth radial dimension as a full-fledged Euclidean spatial dimension, at least so long as we have not observed how any naturally occurring object configurations are best described as 5-polytopes.

One phenomenon which resists explanation in a space of just four dimensions is the propagation of light in a vacuum. The propagation of mass-carrying particles is explained as the consequence of their rotations in closed, curved spaces (3-spheres) of finite size, moving through four-dimensional Euclidean space at a universal constant speed, the speed of light. But an apparent paradox remains that light must seemingly propagate through four-dimensional Euclidean space at more than the speed of light. From a five-dimensional viewpoint, this apparent paradox can be resolved, and in retrospect it is clear how massless particles can translate through four-dimensional space at twice the speed constant, since they are not simultaneously rotating.

Another phenomenon justifying a five-dimensional view of space is the relation between the the 5-cell proton and the 16-cell neutron (the 4-simplex and 4-orthoplex polytopes). Their indirect relationship can be observed in the 4-600-point polytope (the 120-cell), and in its 11-cells,{{Sfn|Christie|2024}} but it is only directly observed (absent a 120-cell) in a five-dimensional reference frame.

=== Nuclear geometry ===
We have seen how isoclinic rotations (Clifford displacements) relate the orbits in the atomic nucleus to each other, just as they relate the regular convex 4-polytopes to each other, in a sequence of nested objects of increasing complexity. We have identified the proton as a  5-point, 5-cell 4-simplex 𝜶&lt;sub&gt;4&lt;/sub&gt;, the neutron as an 8-point, 16-cell 4-orthoplex 𝛽&lt;sub&gt;4&lt;/sub&gt;, and the shell of the atomic nucleus as a 24-point 24-cell. As Coxeter noted, that unique 24-point object stands quite alone in four dimensions, having no analogue above or below.

=== Atomic geometry ===
I'm on a plane flying to Eugene to visit Catalin, we'll talk after I arrive. I've been working on both my unpublished papers, the one going put for pre-publication review soon about 4D geometry, and the big one not going out soon about the 4D sun, 4D atoms, and 4D galaxies and n-D universe. I'vd just added the following paragraph to that big paper:

Atomic geometry

The force binding the protons and neutrons of the nucleus together into a distinct element is specifically an expression of the 11-cell 4-polytope, itself an expression of the pyritohedral symmetry, which binds the distinct 4-polytopes to each other, and relates the n-polytopes to their neighbors of different n by dimensional analogy.


flying over mt shasta out my right-side window at the moment, that last text showing &quot;not delivered&quot; yet because there's no wifi on this plane, gazing at that great peak of the world and feeling as if i've just made the first ascent of it

=== Molecular geometry ===
Molecules are 3-dimensional structures that live in the thin film of 3-membrane only one atom thick in most places that is our ordinary space, but since that is a significantly curved 3-dimensional space at the scale of a molecule, the way the molecule's covalent bonds form is influenced by the local curvature in 4-dimensions at that point.

In the water molecule, there is a reason why the hydrogen atoms are attached to the oxygen atom at an angle of 104.45° in 3-dimensional space, and at root it must be the same symmetry that locates any two of the hydrogen proton's five vertices 104.45° apart on a great circle arc of its tiny 3-sphere.

=== Cosmology ===

==== Solar systems ====

===== Stars =====
...

===== The Kepler problem =====
...

==== Galaxies ====

The spacetime of general relativity is often illustrated as a projection to a curved 2D surface in which large gravitational objects make gravity wells or dimples in the surface. In the Euclidean 4D view of the universe the 3D surface of a large cosmic object such as a galaxy surrounds an empty 4D space, and large gravitational objects within the galaxy must make dimples in its surface. But should we see them as dimples exactly? Would they dimple inwards or outwards? In the spacetime illustrations they are naturally always shown as dimpling downwards, which is somewhat disingenuous, strongly suggesting to the viewer that the reason for gravity is that it flows downhill - the original tautology we are trying to surmount! In the Euclidean 4D galaxy the dimple, if it is one, must be either inward or outward, and which it is matters since the dimple is flying outward at velocity {{mvar|c}}. The galaxy is not collapsing inward. Is a large gravitational mass (such as a star) ''ahead'' of the smaller masses orbiting around it (such as its planets), or is it ''behind'' them, as they fly through 4-space on their Clifford parallel trajectories? The answer is ''both'' of course, because a star is not a dimple, it is a 4-ball, and it dimples the 3D surface both inwards and outwards. It is a thick place in the 3D surface. We should view it as having its gravitational center precisely at the surface of the expanding 3-sphere. 

What is a black hole? It is the hollow four-dimensional space that a galaxy is the three-dimensional surface of.

When we view another galaxy, such as Andromeda, we are seeing that whole galaxy from a distance, the way the moon astronauts looked back at the whole earth. We see our own milky way galaxy from where we are on its surface, the way we see the earth from its surface, except that the earth is solid, but the galaxy is hollow and transparent. We can look across its empty center and see all the other stars also on its surface, including those opposite ours on the far side of its 3-sphere. The thicker band of stars we see in our night sky and identify as the milky way is not our whole galaxy; the majority of the other visible stars also lie in our galaxy. That dense band is not thicker and brighter than other parts of our galaxy because it lies toward a dense galactic center (our galaxy has an empty center), but for exactly the opposite reason: those apparently more thickly clustered stars lie all around us on the galaxy's surface, in the nearest region of space surrounding us. They appear to be densely packed only because we are looking at them &quot;edge on&quot;. Actually, we are looking into this nearby apparently dense region ''face on'', not edge on, because we are looking at a round sphere of space surrounding us, not a disk. In contrast, stars in our galaxy outside that bright band lie farther off from us, across the empty center of the galaxy, and we see them spread out as they actually are, instead of &quot;edge on&quot; so they appear to be densely clustered. The &quot;dense band&quot; covers only an equatorial band of the night sky instead of all the sky, because when we look out into the four-dimensional space around us, we can see stars above and below our three-dimensional hyperplane in our four-dimensional space. Everything in our solar system lies in our hyperplane, and the nearby stars around us in our galaxy are near our hyperplane (just slightly below it). All the other, more distant stars in our galaxy are also below our hyperplane. We can see objects outside our galaxy, such as other galaxies, both above and below our hyperplane. We can see all around us above our hyperplane (looking up from the galactic surface into the fourth dimension), and all around us below our hyperplane (looking down through our transparent galaxy and out the other side).

== Revolutions ==
The original Copernican revolution displaced the center of the universe from the center of the earth to a point farther away, the center of the sun, with the stars remaining on a fixed sphere around the sun instead of around the earth. But this led inevitably to the recognition that the sun must be a star itself, not equidistant from all the stars, and the center of but one of many spheres, no monotheistic center at all. 

In such fashion the Euclidean four-dimensional viewpoint initially lends itself to a big bang theory of a single origin of the whole universe, but leads inevitably to the recognition that all the stars need not be equidistant from a single origin in time, any more than they all lie in the same galaxy, equidistant from its center in space. The expanding sphere of matter on the surface of which we find ourselves living might be one of many such spheres, with their big bang origins occurring at distinct times and places in the 4-dimensional universe.

When we look up at the heavens, we have no obvious way of knowing whether the space we are looking into is a curved 3-spherical one or a flat 4-space. In this work we suggest a theory of how light travels that says we can see into all four dimensions, and so when we look up at night we see cosmological objects distributed in 4-dimensional space, and not all located on our own 3-spherical membrane. The view from our solar system suggests that our galaxy is its own hollow 3-sphere, and that galaxies generally are single roughly spherical 3-membranes, with the smaller objects within them all lying on that same 3-spherical surface, equidistant from the galaxy center in 4-space. 

The Euclidean four-dimensional viewpoint requires that all mass-carrying objects are in motion at constant velocity &lt;math&gt;c&lt;/math&gt;, although the relative velocity between nearby objects is much smaller since they move on similar vectors, aimed away from a common origin point in the past. It is natural to expect that objects moving at constant velocity away from a common origin will be distributed roughly on the surface of an expanding 3-sphere. Since their paths away from their origin are not straight lines but various helical isoclines, their 3-sphere will be expanding radially at slightly less than the constant velocity &lt;math&gt;c&lt;/math&gt;. The view from our solar system does ''not'' suggest that each galaxy is its own distinct 3-sphere expanding at this great rate; rather, the standard theory has been that the entire observable universe is expanding from a single big bang origin in time. While the Euclidean four-dimensional viewpoint lends itself to that standard theory, it also allows theories which require no single origin point in space and time.

These are the voyages of starship Earth, to boldly go where no one has gone before. It made the jump to lightspeed long ago, in whatever big bang its atoms emerged from, and hasn't slowed down since.

== Origins of the theory ==
Einstein himself was one of the first to imagine the universe as the three-dimensional surface of a four-dimensional Euclidean sphere, in what was narrowly the first written articulation of the principle of Euclidean 4-space relativity, contemporaneous with the teen-aged Coxeter's (quoted below). Einstein did this as a [[W:Gedankenexperiment|gedankenexperiment]] in the context of investigating whether his equations of general relativity predicted an infinite or a finite universe, in his 1921 Princeton lecture.&lt;ref&gt;{{Cite book|url=http://www.gutenberg.org/ebooks/36276|title=The Meaning of Relativity|last=Einstein|first=Albert|publisher=Princeton University Press|year=1923|isbn=|location=|pages=110-111}}&lt;/ref&gt; He invited us to imagine &quot;A spherical manifold of three dimensions, embedded in a Euclidean continuum of four dimensions&quot;, but he was careful to disclaim parenthetically that &quot;The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.&quot;

Informally, the Euclidean 4-dimensional theory of relativity may be given as a sort of reciprocal of that formulation of Einstein's: ''The Minkowski spacetime has naturally no significance except that of a mathematical artifice, as an aid to understanding how things will appear to an observer from his perspective; the forthshortenings, clock desynchronizations and other perceptual effects it predicts are exact calculations of actual perspective effects; but space is actually a flat, Euclidean continuum of four orthogonal spatial dimensions, and in it the ordinary laws of a flat vector space hold (such as the Pythagorean theorem), and all sightline calculations work classically, so long as you consider all four dimensions.''

The Euclidean 4-dimensional theory differs from the standard theory in being a description of the physical universe in terms of a geometry of four or more orthogonal spatial dimensions, rather than in the standard theory's terms of the [[w:Minkowski spacetime|Minkowski spacetime]] geometry (in which three spatial dimensions and a time dimension comprise a unified spacetime of four dimensions). The invention of geometry of more than three spatial dimensions preceded Einstein's theories by more than fifty years. It was first worked out by the Swiss mathematician [[w:Ludwig Schläfli|Ludwig Schläfli]] around 1850. Schläfli extended Euclid's geometry of one, two, and three dimensions in a direct way to four or more dimensions, generalizing the rules and terms of [[w:Euclidean geometry|Euclidean geometry]] to spaces of any number of dimensions. He coined the general term ''polyscheme'' to mean geometric forms of any number of dimensions, including two-dimensional [[w:polygon|polygons]], three-dimensional [[w:polyhedron|polyhedra]], four dimensional [[w:polychoron|polychora]], and so on, and in the process he discovered all the [[w:Regular polytope|regular polyschemes]] that are possible in every dimension, including in particular the six convex regular polyschemes which can be constructed in a space of four dimensions (a set analogous to the five [[w:Platonic solid|Platonic solids]] in three dimensional space). Thus he was the first to explore the fourth dimension, reveal its emergent geometric properties, and discover all its astonishing regular objects. Because most of his work remained almost completely unknown until it was published posthumously in 1901, other researchers had more than fifty years to rediscover the regular polyschemes, and competing terms were coined; today [[W:Alicia Boole Stott|Alicia Boole Stott]]'s word ''[[w:Polytope|polytope]]'' is the commonly used term for ''polyscheme''.{{Efn|Today Schläfli's original ''polyscheme'', with its echo of ''schema'' as in the configurations of information structures, seems even more fitting in its generality than ''polytope'' -- perhaps analogously as information software (programming) is even more general than information hardware (computers).}}

== Boundaries ==
&lt;blockquote&gt;Ever since we discovered that Earth is round and turns like a mad-spinning top, we have understood that reality is not as it appears to us: every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.&lt;ref&gt;{{Cite book|author=Carlo Rovelli|title=Seven Brief Lessons on Physics}}&lt;/ref&gt;&lt;/blockquote&gt;

Of course it is strange to consciously contemplate this world we inhabit, our planet, our solar system, our vast galaxy, as the merest film, a boundary no thicker in the places we inhabit than the diameter of an electron (though much thicker in some places we cannot inhabit, such as the interior of stars). But is not our unconscious traditional concept of the boundary of our world even stranger? Since the enlightenment we are accustomed to thinking that there is nothing beyond three dimensional space: no boundary, because there is nothing else to separate us from. But anyone who knows the [[polyscheme]]s Schlafli discovered knows that space can have any number of dimensions, and that there are fundamental objects and motions to be discovered in four dimensions that are even more various and interesting than those we can discover in three. The strange thing, when we think about it, is that there ''is'' a boundary between three and four dimensions. ''Why'' can't we move (or apparently, see) in more than three dimensions? Why is our world apparently only three dimensional? Why would it have ''three'' dimensions, and not four, or five, or the ''n'' dimensions that Schlafli mapped? What is the nature of the boundary which confines us to just three? 

We know that in Euclidean geometry the boundary between three and four dimensions is itself a spherical three dimensional space, so we should suspect that we are materially confined within such a curved boundary. Light need not be confined with us within our three dimensional boundary space. We would look directly through four dimensional space in our natural way by receiving light signals that traveled to us on straight lines through it. The reason we do not observe a fourth spatial dimension in our vicinity is that there are no nearby objects in it, just off our hyperplane in the wild. The nearest four-dimensional object we can see with our eyes is our sun, which lies equatorially in our own hyperplane, though it bulges out of it above and below. But when we look up at the heavens, every pinprick of light we observe is itself a four-dimensional object off our hyperplane, and they are distributed around us in four-dimensional space through which we gaze. We are four-dimensionally sighted creates, even though our bodies are three-dimensional objects, thin as an atom in the fourth dimension. But that should not surprise us: we can see into three dimensional space even though our retinas are two dimensional objects, thin as a photoreceptor cell.

Our unconscious provincial concept is that there is nothing else outside our three dimensional world: no boundary, because there is nothing else to separate us from. But Schlafli discovered something else: all the astonishing regular objects that exist in higher dimensions. So this conception now has the same kind of status as our idea that the sun rises in the east and passes overhead: it is mere appearance, not a true model and not a proper explanation. A boundary is an explanation, be it ever so thin. And would a boundary of ''no'' thickness, a mere abstraction with no physical power to separate, be a more suitable explanation?

&lt;blockquote&gt;The number of dimensions possessed by a figure is the number of straight lines each perpendicular to all the others which can be drawn on it. Thus a point has no dimensions, a straight line one, a plane surface two, and a solid three ....

In space as we now know it only three lines can be imagined perpendicular to each other. A fourth line, perpendicular to all the other three would be quite invisible and unimaginable to us. We ourselves and all the material things around us probably possess a fourth dimension, of which we are quite unaware. If not, from a four-dimensional point of view we are mere geometrical abstractions, like geometrical surfaces, lines, and points are to us. But this thickness in the fourth dimension must be exceedingly minute, if it exists at all. That is, we could only draw an exceedingly small line perpendicular to our three perpendicular lines, length, breadth and thickness, so small that no microscope could ever perceive it.

We can find out something about the conditions of the fourth and higher dimensions if they exist, without being certain that they do exist, by a process which I have termed &quot;Dimensional Analogy.&quot;&lt;ref&gt;{{Citation|title=Dimensional Analogy|last=Coxeter|first=Donald|date=February 1923|publisher=Coxeter Fonds, University of Toronto Archives|authorlink=W:Harold Scott MacDonald Coxeter|series=|postscript=|work=}}&lt;/ref&gt;&lt;/blockquote&gt;

I believe, but I cannot prove, that our universe is properly a Euclidean space of four orthogonal spatial dimensions. Others will have to work out the physics and do the math, because I don't have the mathematics; entirely unlike Coxeter and Einstein, I am illiterate in those languages.

&lt;blockquote&gt;
::::::BEECH

:Where my imaginary line
:Bends square in woods, an iron spine
:And pile of real rocks have been founded.
:And off this corner in the wild,
:Where these are driven in and piled,
:One tree, by being deeply wounded,
:Has been impressed as Witness Tree
:And made commit to memory
:My proof of being not unbounded.
:Thus truth's established and borne out,
:Though circumstanced with dark and doubt—
:Though by a world of doubt surrounded.

:::::::—''The Moodie Forester''&lt;ref&gt;{{Cite book|title=A Witness Tree|last=Frost|first=Robert|year=1942|series=The Poetry of Robert Frost|publisher=Holt, Rinehart and Winston|edition=1969|}}&lt;/ref&gt;
&lt;/blockquote&gt;

== Sequence of regular 4-polytopes ==
{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}|columns=9}}

== Notes ==
{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions (two completely orthogonal planes) at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle. A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways. All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:cubic pyramid]|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}}

{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The vertex cubic pyramid is not actually radially equilateral,{{Efn|name=radially equilateral}} because the edges radiating from its apex are not actually its radii: the apex of the [[W:cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices.}}|name=24-cell vertex figure}}

{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,–&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,–1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}}

{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[24-cell#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[24-cell#Hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[24-cell#Squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:tesseract|hypercube (a tesseract or 8-cell)]], in [[24-cell#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells. The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[24-cell#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[24-cell#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[3-sphere]] symmetric: four [[24-cell#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' completely orthogonal directions at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart as in a simple rotation.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell by rotating toward it; it can only reach the 16-cell ''beyond'' it. But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only sense in which the two 16-cells are left or right of each other.)|name=three isoclinic 16-cells}}

{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in the plane ''as the plane moves'', tilting sideways by the same angle that the other plane rotates.|name=helical geodesic}}

{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''. A '''[[W:geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:screw threads|screw threads]] either, because they form a closed loop like any circle (after ''two'' revolutions). Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in a circle in two completely orthogonal directions at once.{{Efn|Isoclinic geodesics are ''4-dimensional great circles'' in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two completely orthogonal planes at once. They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of 2-dimensional great circles (great 1-spheres).}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are geodesics, and isoclines on the 3-sphere are 4-dimensionally circular, but not all isoclines on 3-manifolds in 4-space are perfectly circular.}} they always occur in [[W:chiral|chiral]] pairs and form a pair of [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{Efn|Isoclines on the 3-sphere occur in non-intersecting chiral pairs. A left and a right isocline form a [[W:Hopf link|Hopf link]] called the {1,1} torus knot{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also
known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two
intersecting circles that are the cross-section of a torus by a well-chosen plane
cutting it. Picking one such circle and rotating it around the torus
axis, the resulting family of circles can be used to rule the torus. By nesting
tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the
(1, 1) torus knot [a [[W:Hopf link|Hopf link]]] rather than as a planar cut [two intersecting circles].&quot;}} in which ''each'' of the two linked circles traverses all four dimensions.}} the paths of the left and the right [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] twice around the 3-sphere through the non-adjacent vertices of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew polygon]].|name=isoclinic geodesic}}

{{Efn|[[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center, the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}}

{{Efn|In the 24-cell each great square plane is completely orthogonal{{Efn|name=completely orthogonal planes}} to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two vertices: a great [[W:digon|digon]] plane.|name=pairs of completely orthogonal planes}}

{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance]] equal to the square root of four times the square of that distance. For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} all vertices are displaced to a vertex two edge lengths away. Each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} in four orthogonal coordinate directions.|name=isoclinic 4-dimensional diagonal}}

{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but completely orthogonal{{Efn|name=completely orthogonal planes}} to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}}

{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}}

{{Efn|In 4 dimensional space we can construct 4 perpendicular axes and 6 perpendicular planes through a point. Without loss of generality, we may take these to be the axes and orthogonal central planes of a (w, x, y, z) Cartesian coordinate system. In 4 dimensions we have the same 3 orthogonal planes (xy, xz, yz) that we have in 3 dimensions, and also 3 others (wx, wy, wz). Each of the 6 orthogonal planes shares an axis with 4 of the others, and is ''completely orthogonal'' to just one of the others: the only one with which it does not share an axis. Thus there are 3 pairs of completely orthogonal planes: xy and wz intersect only at the origin; xz and wy intersect only at the origin; yz and wx intersect only at the origin.|name=six orthogonal planes of the Cartesian basis}}

{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} (and they ''must'', if they are completely orthogonal).{{Efn|Two flat planes A and B of a Euclidean space of four dimensions are called ''completely orthogonal'' if and only if every line in A is orthogonal to every line in B. In that case the planes A and B intersect at a single point O, so that if a line in A intersects with a line in B, they intersect at O.{{Efn|name=six orthogonal planes of the Cartesian basis}}|name=completely orthogonal planes}}|name=how planes intersect}}

{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}}

{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;}} Since all planes in the same [[W:hyperplane|hyperplane]] are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles (completely orthogonal){{Efn|name=completely orthogonal planes}} or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes are neither isoclinic nor Clifford parallel; they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}}

{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diagonals). The 8-cells are not completely disjoint{{Efn|name=completely disjoint}} (they share vertices), but each cube and each {{radic|3}} chord belongs to just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell.|name=three 8-cells}}

{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline. More generally, isoclines are geodesics because the distance between their ''adjacent'' vertices is the shortest distance between those two vertices, but a path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}}, P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation.|name=360 degree geodesic path visiting 3 hexagonal planes}}

{{Notelist|40em}}

== Citations ==
{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}}

{{Reflist|40em}}

== References ==
{{Refbegin}}
* {{Cite book | last=Kepler | first=Johannes | author-link=W:Johannes Kepler | title=Harmonices Mundi (The Harmony of the World) | title-link=W:Harmonices Mundi | publisher=Johann Planck | year=1619}}
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** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions''
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* {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}}
* {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }}
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* {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}}
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* {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|url=https://doi.org/10.1007/s00006-019-0960-5}}
* {{Cite journal|title=Theoretical Evidence for Principles of Special Relativity Based on Isotropic and Uniform Four-Dimensional Space|first=Takuya|last=Yamashita|date=25 May 2023|doi= 10.20944/preprints202305.1785.v1|journal=Preprints|volume=2023|issue=2023051785|url=https://doi.org/10.20944/preprints202305.1785.v1}}
*{{Citation | last=Goucher | first=A.P. | title=Spin groups | date=19 November 2019 | journal=Complex Projective 4-Space | url=https://cp4space.hatsya.com/2012/11/19/spin-groups/ }}
* {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=A symmetrical arrangement of 120 11-cells|title-link=User:Dc.samizdat/A symmetrical arrangement of 120 11-cells|journal=Wikiversity}}
{{Refend}}</text>
      <sha1>aerwj4o6gqa1pajghm4reu8ayujo4g9</sha1>
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  <page>
    <title>24-cell</title>
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      <timestamp>2024-12-15T22:39:56Z</timestamp>
      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Rotations */ cite Goucher 2019 on Spin Groups</comment>
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      <text bytes="247568" sha1="3fahin09p1rcjgh4ebtn53uwhftvwlq" xml:space="preserve">{{Short description|Regular object in four dimensional geometry}}
{{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}}
{{Infobox 4-polytope
 | Name=24-cell
 | Image_File=Schlegel wireframe 24-cell.png
 | Image_Caption=[[W:Schlegel diagram|Schlegel diagram]]&lt;br&gt;(vertices and edges)
 | Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]
 | Last=[[W:Omnitruncated tesseract|21]]
 | Index=22
 | Next=[[W:Rectified 24-cell|23]]
 | Schläfli={3,4,3}&lt;br&gt;r{3,3,4} = &lt;math&gt;\left\{\begin{array}{l}3\\3,4\end{array}\right\}&lt;/math&gt;&lt;br&gt;{3&lt;sup&gt;1,1,1&lt;/sup&gt;} = &lt;math&gt;\left\{\begin{array}{l}3\\3\\3\end{array}\right\}&lt;/math&gt;
 | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}&lt;br&gt;{{Coxeter–Dynkin diagram|node|3|node_1|split1|nodes}} or {{Coxeter–Dynkin diagram|node_1|splitsplit1|branch3|node}}
 | Cell_List=24 [[W:Octahedron|{3,4}]] [[File:Octahedron.png|20px]]
 | Face_List=96 [[W:Triangle|{3}]]
 | Edge_Count=96
 | Vertex_Count= 24
 | Petrie_Polygon=[[W:Dodecagon|{12}]]
 | Coxeter_Group=[[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]], [3,4,3], order 1152&lt;br&gt;B&lt;sub&gt;4&lt;/sub&gt;, [4,3,3], order 384&lt;br&gt;D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
 | Vertex_Figure=[[W:Cube|cube]]
 | Dual=[[W:Polytope#Self-dual polytopes|self-dual]]
 | Property_List=[[W:Convex polytope|convex]], [[W:Isogonal figure|isogonal]], [[W:Isotoxal figure|isotoxal]], [[W:Isohedral figure|isohedral]]
}}
[[File:24-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]]
In [[W:four-dimensional space|four-dimensional geometry]], the '''24-cell''' is the convex [[W:Regular 4-polytope|regular 4-polytope]]{{Sfn|Coxeter|1973|p=118|loc=Chapter VII: Ordinary Polytopes in Higher Space}} (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,4,3}. It is also called '''C&lt;sub&gt;24&lt;/sub&gt;''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for &quot;octahedral complex&quot;), '''icosatetrahedroid''',{{sfn|Ghyka|1977|p=68}} '''[[W:Octacube (sculpture)|octacube]]''', '''hyper-diamond''' or '''polyoctahedron''', being constructed of [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].

The boundary of the 24-cell is composed of 24 [[W:Octahedron|octahedral]] cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The [[W:Vertex figure|vertex figure]] is a [[W:Cube|cube]]. The 24-cell is [[W:Self-dual polyhedron|self-dual]].{{Efn|The 24-cell is one of only three self-dual regular Euclidean polytopes which are neither a [[W:Polygon|polygon]] nor a [[W:Simplex|simplex]]. The other two are also 4-polytopes, but not convex: the [[W:Grand stellated 120-cell|grand stellated 120-cell]] and the [[W:Great 120-cell|great 120-cell]]. The 24-cell is nearly unique among self-dual regular convex polytopes in that it and the even polygons are the only such polytopes where a face is not opposite an edge.|name=|group=}} The 24-cell and the [[W:Tesseract|tesseract]] are the only convex regular 4-polytopes in which the edge length equals the radius.{{Efn||name=radially equilateral|group=}}

The 24-cell does not have a regular analogue in [[W:Three dimensions|three dimensions]] or any other number of dimensions, either below or above.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

Translated copies of the 24-cell can [[W:Tesselate|tesselate]] four-dimensional space face-to-face, forming the [[W:24-cell honeycomb|24-cell honeycomb]]. As a polytope that can tile by translation, the 24-cell is an example of a [[W:Parallelohedron|parallelotope]], the simplest one that is not also a [[W:Zonotope|zonotope]].{{Sfn|Coxeter|1968|p=70|loc=§4.12 The Classification of Zonohedra}}

==Geometry==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s (in order of size and complexity).{{Efn|name=4-polytopes ordered by size and complexity}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} It can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|tesseract]] (8-cell), as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.{{Efn|name=edge length of successor}}

=== Coordinates ===
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

==== Great squares ====
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]]. Such polytopes are ''radially equilateral''.{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius={{radic|2}}}}

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]] great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

==== Great hexagons ====
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[#Quaternionic interpretation|Viewed as quaternions]],{{Efn|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]].

The 24-cell has unit radius and unit edge length{{Efn||name=radially equilateral}} in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used [[#Great squares|above]].{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

==== Triangles ====
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares.{{Efn|The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms a tesseract (8-cell).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.|name=great linking triangles}}

==== Hypercubic chords ====
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn||name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

==== Geodesics ====
[[Image:stereographic polytope 24cell faces.png|thumb|[[W:Stereographic projection|Stereographic projection]] of the 24-cell's 16 central hexagons onto their great circles. Each great circle is divided into 6 arc-edges at the intersections where 4 great circles cross.]]
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell,{{Efn|name=radially equilateral}} and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

{| class=&quot;wikitable floatright&quot;
|+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell
|- style=&quot;text-align:center;&quot;
![[W:Coxeter plane|Coxeter plane]]
!colspan=2|F&lt;sub&gt;4&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|colspan=2|[[File:24-cell t0_F4.svg|100px]]
|- style=&quot;text-align:center;&quot;
![[W:Dihedral symmetry|Dihedral symmetry]]
|colspan=2|[12]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (a)
!B&lt;sub&gt;3&lt;/sub&gt; / A&lt;sub&gt;2&lt;/sub&gt; (b)
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B3.svg|100px]]
|[[File:24-cell t3_B3.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[6]
|[6]
|- style=&quot;text-align:center;&quot;
!Coxeter plane
!B&lt;sub&gt;4&lt;/sub&gt;
!B&lt;sub&gt;2&lt;/sub&gt; / A&lt;sub&gt;3&lt;/sub&gt;
|- style=&quot;text-align:center;&quot;
!Graph
|[[File:24-cell t0_B4.svg|100px]]
|[[File:24-cell t0_B2.svg|100px]]
|- style=&quot;text-align:center;&quot;
!Dihedral symmetry
|[8]
|[4]
|}
The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|name=hyperplanes}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

=== Constructions ===
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#As a configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The 600-cell is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular 5-cell is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|tesseract]] (8-cell).{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge.{{Efn|name=radially equilateral|group=}} They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

==== Relationships among interior polytopes ====
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.{{Efn|name=great linking triangles}}[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

==== Boundary cells ====
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

=== As a configuration ===
This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 24-cell. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers say how many of each element occur in the whole 24-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element.

&lt;math display=&quot;block&quot;&gt;\begin{bmatrix}\begin{matrix}24 &amp; 8 &amp; 12 &amp; 6 \\ 2 &amp; 96 &amp; 3 &amp; 3 \\ 3 &amp; 3 &amp; 96 &amp; 2 \\ 6 &amp; 12 &amp; 8 &amp; 24 \end{matrix}\end{bmatrix}&lt;/math&gt;

Since the 24-cell is self-dual, its matrix is identical to its 180 degree rotation.

==Symmetries, root systems, and tessellations==
[[File:F4 roots by 24-cell duals.svg|thumb|upright|The compound of the 24 vertices of the 24-cell (red nodes), and its unscaled dual (yellow nodes), represent the 48 root vectors of the [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]] group, as shown in this F&lt;sub&gt;4&lt;/sub&gt; Coxeter plane projection]]

The 24 root vectors of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt; root system]] of the [[W:Simple Lie group|simple Lie group]] [[W:SO(8)|SO(8)]] form the vertices of a 24-cell. The vertices can be seen in 3 [[W:Hyperplane|hyperplane]]s,{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} with the 6 vertices of an [[W:Octahedron|octahedron]] cell on each of the outer hyperplanes and 12 vertices of a [[W:Cuboctahedron|cuboctahedron]] on a central hyperplane. These vertices, combined with the 8 vertices of the [[16-cell]], represent the 32 root vectors of the B&lt;sub&gt;4&lt;/sub&gt; and C&lt;sub&gt;4&lt;/sub&gt; simple Lie groups.

The 48 vertices (or strictly speaking their radius vectors) of the union of the 24-cell and its dual form the [[W:Root system|root system]] of type [[W:F4 (mathematics)|F&lt;sub&gt;4&lt;/sub&gt;]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D&lt;sub&gt;4&lt;/sub&gt;; its size has the ratio {{sqrt|2}}:1. This is likewise true for the 24 vertices of its dual. The full [[W:Symmetry group|symmetry group]] of the 24-cell is the [[W:Weyl group|Weyl group]] of F&lt;sub&gt;4&lt;/sub&gt;, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F&lt;sub&gt;4&lt;/sub&gt; roots. This is a [[W:Solvable group|solvable group]] of order 1152. The rotational symmetry group of the 24-cell is of order 576.

===Quaternionic interpretation===
[[File:Binary tetrahedral group elements.png|thumb|The 24 quaternion{{Efn|name=quaternions}} elements of the [[W:Binary tetrahedral group|binary tetrahedral group]] match the vertices of the 24-cell. Seen in 4-fold symmetry projection:
* 1 order-1: 1
* 1 order-2: -1
* 6 order-4: ±i, ±j, ±k
* 8 order-6: (+1±i±j±k)/2
* 8 order-3: (-1±i±j±k)/2.]]When interpreted as the [[W:Quaternion|quaternion]]s,{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} the F&lt;sub&gt;4&lt;/sub&gt; [[W:root lattice|root lattice]] (which is the integral span of the vertices of the 24-cell) is closed under multiplication and is therefore a [[W:ring (mathematics)|ring]]. This is the ring of [[W:Hurwitz integral quaternion|Hurwitz integral quaternion]]s. The vertices of the 24-cell form the [[W:Group of units|group of units]] (i.e. the group of invertible elements) in the Hurwitz quaternion ring (this group is also known as the [[W:Binary tetrahedral group|binary tetrahedral group]]). The vertices of the 24-cell are precisely the 24 Hurwitz quaternions with norm squared 1, and the vertices of the dual 24-cell are those with norm squared 2. The D&lt;sub&gt;4&lt;/sub&gt; root lattice is the [[W:Dual lattice|dual]] of the F&lt;sub&gt;4&lt;/sub&gt; and is given by the subring of Hurwitz quaternions with even norm squared.{{Sfn|Egan|2021|ps=; quaternions, the binary tetrahedral group and the binary octahedral group, with rotating illustrations.}}

Viewed as the 24 unit [[W:Hurwitz quaternion|Hurwitz quaternion]]s, the [[#Great hexagons|unit radius coordinates]] of the 24-cell represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

Vertices of other [[W:Convex regular 4-polytope|convex regular 4-polytope]]s also form multiplicative groups of quaternions, but few of them generate a root lattice.{{Sfn|Koca|Al-Ajmi|Koc|2007}}

===Voronoi cells===
The [[W:Voronoi cell|Voronoi cell]]s of the [[W:D4 (root system)|D&lt;sub&gt;4&lt;/sub&gt;]] root lattice are regular 24-cells. The corresponding Voronoi tessellation gives the [[W:Tessellation|tessellation]] of 4-dimensional [[W:Euclidean space|Euclidean space]] by regular 24-cells, the [[W:24-cell honeycomb|24-cell honeycomb]]. The 24-cells are centered at the D&lt;sub&gt;4&lt;/sub&gt; lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F&lt;sub&gt;4&lt;/sub&gt; lattice points with odd norm squared. Each 24-cell of this tessellation has 24 neighbors. With each of these it shares an octahedron. It also has 24 other neighbors with which it shares only a single vertex. Eight 24-cells meet at any given vertex in this tessellation. The [[W:Schläfli symbol|Schläfli symbol]] for this tessellation is {3,4,3,3}. It is one of only three regular tessellations of '''R'''&lt;sup&gt;4&lt;/sup&gt;.

The unit [[W:Ball (mathematics)|balls]] inscribed in the 24-cells of this tessellation give rise to the densest known [[W:lattice packing|lattice packing]] of [[W:Hypersphere|hypersphere]]s in 4 dimensions. The vertex configuration of the 24-cell has also been shown to give the [[W:24-cell honeycomb#Kissing number|highest possible kissing number in 4 dimensions]].

===Radially equilateral honeycomb===
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.{{Efn||name=radially equilateral}}

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other. Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
The [[#Geometry|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Efn|name=planes through vertices}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]] to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope{{Efn|name=radially equilateral}} is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of non-intersecting linked great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 non-intersecting linked great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of this article. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical hexagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram&lt;sub&gt;2&lt;/sub&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center,{{Efn|name=radially equilateral}} this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
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|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
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|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}

== Visualization ==
[[File:OctacCrop.jpg|thumb|[[W:Octacube (sculpture)|Octacube steel sculpture]] at Pennsylvania State University]]

=== Cell rings ===
The 24-cell is bounded by 24 [[W:Octahedron|octahedral]] [[W:Cell (geometry)|cells]].  For visualization purposes, it is convenient that the octahedron has opposing parallel [[W:Face (geometry)|faces]] (a trait it shares with the cells of the [[W:Tesseract|tesseract]] and the [[120-cell]]).  One can stack octahedrons face to face in a straight line bent in the 4th direction into a [[W:Great circle|great circle]] with a [[W:Circumference|circumference]] of 6 cells.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} The cell locations lend themselves to a [[W:3-sphere|hyperspherical]] description. Pick an arbitrary cell and label it the &quot;[[W:North Pole|North Pole]]&quot;. Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd &quot;[[W:South Pole|South Pole]]&quot; cell. This skeleton accounts for 18 of the 24 cells (2&amp;nbsp;+&amp;nbsp;{{gaps|8|×|2}}). See the table below.

There is another related [[#Geodesics|great circle]] in the 24-cell, the dual of the one above. A path that traverses 6 vertices solely along edges resides in the dual of this polytope, which is itself since it is self dual. These are the [[#Great hexagons|hexagonal]] geodesics [[#Geodesics|described above]].{{Efn|name=hexagonal fibrations}} One can easily follow this path in a rendering of the equatorial [[W:Cuboctahedron|cuboctahedron]] cross-section.

Starting at the North Pole, we can build up the 24-cell in 5 latitudinal layers. With the exception of the poles, each layer represents a separate 2-sphere, with the equator being a great 2-sphere.{{Efn|name=great 2-spheres}} The cells labeled equatorial in the following table are interstitial to the meridian great circle cells. The interstitial &quot;equatorial&quot; cells touch the meridian cells at their faces. They touch each other, and the pole cells at their vertices.  This latter subset of eight non-meridian and pole cells has the same relative position to each other as the cells in a [[W:Tesseract|tesseract]] (8-cell), although they touch at their vertices instead of their faces.

{| class=&quot;wikitable&quot;
|-
! Layer #
! Number of Cells
! Description
! Colatitude
! Region
|-
| style=&quot;text-align: center&quot; | 1
| style=&quot;text-align: center&quot; | 1 cell
| North Pole
| style=&quot;text-align: center&quot; | 0°
| rowspan=&quot;2&quot; | Northern Hemisphere
|-
| style=&quot;text-align: center&quot; | 2
| style=&quot;text-align: center&quot; | 8 cells
| First layer of meridian cells
| style=&quot;text-align: center&quot; | 60°
|-
| style=&quot;text-align: center&quot; | 3
| style=&quot;text-align: center&quot; | 6 cells
| Non-meridian / interstitial
| style=&quot;text-align: center&quot; | 90°
| style=&quot;text-align: center&quot; |Equator
|-
| style=&quot;text-align: center&quot; | 4
| style=&quot;text-align: center&quot; | 8 cells
| Second layer of meridian cells
| style=&quot;text-align: center&quot; | 120°
| rowspan=&quot;2&quot; | Southern Hemisphere
|-
| style=&quot;text-align: center&quot; | 5
| style=&quot;text-align: center&quot; | 1 cell
| South Pole
| style=&quot;text-align: center&quot; | 180°
|-
! Total
! 24 cells
! colspan=&quot;3&quot; |
|}
[[File:24-cell-6 ring edge center perspective.png|thumb|An edge-center perspective projection, showing one of four rings of 6 octahedra around the equator]]
The 24-cell can be partitioned into cell-disjoint sets of four of these 6-cell great circle rings, forming a discrete [[W:Hopf fibration|Hopf fibration]] of four non-intersecting linked rings.{{Efn|name=fibrations are distinguished only by rotations}} One ring is &quot;vertical&quot;, encompassing the pole cells and four meridian cells. The other three rings each encompass two equatorial cells and four meridian cells, two from the northern hemisphere and two from the southern.{{sfn|Banchoff|2013|p=|pp=265-266|loc=}}

Note this hexagon great circle path implies the interior/dihedral angle between adjacent cells is 180 - 360/6 = 120 degrees.  This suggests you can adjacently stack exactly three 24-cells in a plane and form a 4-D honeycomb of 24-cells as described previously.

One can also follow a [[#Geodesics|great circle]] route, through the octahedrons' opposing vertices, that is four cells long.  These are the [[#Great squares|square]] geodesics along four {{sqrt|2}} chords [[#Geodesics|described above]]. This path corresponds to traversing diagonally through the squares in the cuboctahedron cross-section.  The 24-cell is the only regular polytope in more than two dimensions where you can traverse a great circle purely through opposing vertices (and the interior) of each cell.  This great circle is self dual.  This path was touched on above regarding the set of 8 non-meridian (equatorial) and pole cells.

The 24-cell can be equipartitioned into three 8-cell subsets, each having the organization of a tesseract.  Each of these subsets can be further equipartitioned into two non-intersecting linked great circle chains, four cells long.  Collectively these three subsets now produce another, six ring, discrete Hopf fibration.

=== Parallel projections ===
[[Image:Orthogonal projection envelopes 24-cell.png|thumb|Projection envelopes of the 24-cell. (Each cell is drawn with different colored faces, inverted cells are undrawn)]]
The ''vertex-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Rhombic dodecahedron|rhombic dodecahedral]] [[W:Projection envelope|envelope]]. Twelve of the 24 octahedral cells project in pairs onto six square dipyramids that meet at the center of the rhombic dodecahedron. The remaining 12 octahedral cells project onto the 12 rhombic faces of the rhombic dodecahedron.

The ''cell-first'' parallel projection of the 24-cell into 3-dimensional space has a [[W:Cuboctahedron|cuboctahedral]] envelope. Two of the octahedral cells, the nearest and farther from the viewer along the ''w''-axis, project onto an octahedron whose vertices lie at the center of the cuboctahedron's square faces. Surrounding this central octahedron lie the projections of 16 other cells, having 8 pairs that each project to one of the 8 volumes lying between a triangular face of the central octahedron and the closest triangular face of the cuboctahedron. The remaining 6 cells project onto the square faces of the cuboctahedron. This corresponds with the decomposition of the cuboctahedron into a regular octahedron and 8 irregular but equal octahedra, each of which is in the shape of the convex hull of a cube with two opposite vertices removed.

The ''edge-first'' parallel projection has an [[W:Elongated hexagonal dipyramidelongated hexagonal dipyramid|Elongated hexagonal dipyramidelongated hexagonal dipyramid]]al envelope, and the ''face-first'' parallel projection has a nonuniform hexagonal bi-[[W:Hexagonal antiprism|antiprismic]] envelope.

=== Perspective projections ===
The ''vertex-first'' [[W:Perspective projection|perspective projection]] of the 24-cell into 3-dimensional space has a [[W:Tetrakis hexahedron|tetrakis hexahedral]] envelope. The layout of cells in this image is similar to the image under parallel projection.

The following sequence of images shows the structure of the cell-first perspective projection of the 24-cell into 3 dimensions. The 4D viewpoint is placed at a distance of five times the vertex-center radius of the 24-cell.

{|class=&quot;wikitable&quot; width=660
!colspan=3|Cell-first perspective projection
|- valign=top
|[[Image:24cell-perspective-cell-first-01.png|220px]]&lt;BR&gt;In this image, the nearest cell is rendered in red, and the remaining cells are in edge-outline. For clarity, cells facing away from the 4D viewpoint have been culled.
|[[Image:24cell-perspective-cell-first-02.png|220px]]&lt;BR&gt;In this image, four of the 8 cells surrounding the nearest cell are shown in green. The fourth cell is behind the central cell in this viewpoint (slightly discernible since the red cell is semi-transparent).
|[[Image:24cell-perspective-cell-first-03.png|220px]]&lt;BR&gt;Finally, all 8 cells surrounding the nearest cell are shown, with the last four rendered in magenta.
|-
|colspan=3|Note that these images do not include cells which are facing away from the 4D viewpoint. Hence, only 9 cells are shown here. On the far side of the 24-cell are another 9 cells in an identical arrangement. The remaining 6 cells lie on the &quot;equator&quot; of the 24-cell, and bridge the two sets of cells.
|}

{| class=&quot;wikitable&quot; width=440
|[[Image:24cell section anim.gif|220px]]&lt;br&gt;Animated cross-section of 24-cell
|-
|colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]&lt;br&gt;A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell).
|-
|colspan=3|[[File:Cell24Construction.ogv|450px]]&lt;br&gt;Isometric Orthogonal Projection of: 8 Cell(Tesseract) + 16 Cell = 24 Cell
|}

== Related polytopes ==
=== Three Coxeter group constructions ===
There are two lower symmetry forms of the 24-cell, derived as a [[W:Rectification (geometry)|rectified]] 16-cell, with B&lt;sub&gt;4&lt;/sub&gt; or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D&lt;sub&gt;4&lt;/sub&gt; or [3&lt;sup&gt;1,1,1&lt;/sup&gt;] symmetry, and drawn tricolored with 8 octahedra each.&lt;!-- it would be nice to illustrate another of these lower-symmetry decompositions of the 24-cell, into 4 different-colored helixes of 6 face-bonded octahedral cells, as those are the cell rings of its fibration described in /* Visualization */ --&gt;

{| class=&quot;wikitable collapsible collapsed&quot;
!colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D&lt;sub&gt;4&lt;/sub&gt;, B&lt;sub&gt;4&lt;/sub&gt;, and F&lt;sub&gt;4&lt;/sub&gt; symmetry
|-
![[W:Rectified demitesseract|Rectified demitesseract]]
![[W:Rectified demitesseract|Rectified 16-cell]]
!Regular 24-cell
|-
!D&lt;sub&gt;4&lt;/sub&gt;, [3&lt;sup&gt;1,1,1&lt;/sup&gt;], order 192
!B&lt;sub&gt;4&lt;/sub&gt;, [3,3,4], order 384
!F&lt;sub&gt;4&lt;/sub&gt;, [3,4,3], order 1152
|-
|colspan=3 align=center|[[Image:24-cell net 3-symmetries.png|659px]]
|- valign=top
|width=213|Three sets of 8 [[W:Rectified tetrahedron|rectified tetrahedral]] cells
|width=213|One set of 16 [[W:Rectified tetrahedron|rectified tetrahedral]] cells and one set of 8 [[W:Octahedron|octahedral]] cells.
|width=213|One set of 24 [[W:Octahedron|octahedral]] cells
|-
|colspan=3 align=center|'''[[W:Vertex figure|Vertex figure]]'''&lt;br&gt;(Each edge corresponds to one triangular face, colored by symmetry arrangement)
|- align=center
|[[Image:Rectified demitesseract verf.png|120px]]
|[[Image:Rectified 16-cell verf.png|120px]]
|[[Image:24 cell verf.svg|120px]]
|}

=== Related complex polygons ===
The [[W:Regular complex polygon|regular complex polygon]] &lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} or {{Coxeter–Dynkin diagram|node_h|6|4node}} contains the 24 vertices of the 24-cell, and 24 4-edges that correspond to central squares of 24 of 48 octahedral cells. Its symmetry is &lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, order 96.{{Sfn|Coxeter|1991|p=}}

The regular complex polytope &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in &lt;math&gt;\mathbb{C}^2&lt;/math&gt; has a real representation as a 24-cell in 4-dimensional space. &lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; has 24 vertices, and 24 3-edges. Its symmetry is &lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, order 72.

{| class=wikitable width=600
|+ Related figures in orthogonal projections
|-
!Name
!{3,4,3}, {{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}
!&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}}
!&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node_1|4|3node}}
|-
!Symmetry
![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152
!&lt;sub&gt;4&lt;/sub&gt;[3]&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96
!&lt;sub&gt;3&lt;/sub&gt;[4]&lt;sub&gt;3&lt;/sub&gt;, {{Coxeter–Dynkin diagram|3node|4|3node}}, order 72
|- align=center
!Vertices
|24||24||24
|- align=center
!Edges
|96 2-edges||24 4-edge||24 3-edges
|- valign=top
!valign=center|Image
|[[File:24-cell t0 F4.svg|200px]]&lt;BR&gt;24-cell in F4 Coxeter plane, with 24 vertices in two rings of 12, and 96 edges.
|[[File:Complex polygon 4-3-4.png|200px]]&lt;BR&gt;&lt;sub&gt;4&lt;/sub&gt;{3}&lt;sub&gt;4&lt;/sub&gt;, {{Coxeter–Dynkin diagram|4node_1|3|4node}} has 24 vertices and 32 4-edges, shown here with 8 red, green, blue, and yellow square 4-edges.
|[[File:Complex polygon 3-4-3-fill1.png|200px]]&lt;BR&gt;&lt;sub&gt;3&lt;/sub&gt;{4}&lt;sub&gt;3&lt;/sub&gt; or {{Coxeter–Dynkin diagram|3node_1|4|3node}} has 24 vertices and 24 3-edges, shown here with 8 red, 8 green, and 8 blue square 3-edges, with blue edges filled.
|}

=== Related 4-polytopes ===
Several [[W:Uniform 4-polytope|uniform 4-polytope]]s can be derived from the 24-cell via [[W:Truncation (geometry)|truncation]]:
* truncating at 1/3 of the edge length yields the [[W:Truncated 24-cell|truncated 24-cell]];
* truncating at 1/2 of the edge length yields the [[W:Rectified 24-cell|rectified 24-cell]];
* and truncating at half the depth to the dual 24-cell yields the [[W:Bitruncated 24-cell|bitruncated 24-cell]], which is [[W:Cell-transitive|cell-transitive]].

The 96 edges of the 24-cell can be partitioned into the [[W:Golden ratio|golden ratio]] to produce the 96 vertices of the [[W:Snub 24-cell|snub 24-cell]]. This is done by first placing vectors along the 24-cell's edges such that each two-dimensional face is bounded by a cycle, then similarly partitioning each edge into the golden ratio along the direction of its vector. An analogous modification to an [[W:Octahedron|octahedron]] produces an [[W:Regular icosahedron|icosahedron]], or &quot;[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]].&quot;

The 24-cell is the unique convex self-dual regular Euclidean polytope that is neither a [[W:Polygon|polygon]] nor a [[W:simplex (geometry)|simplex]]. Relaxing the condition of convexity admits two further figures: the [[W:Great 120-cell|great 120-cell]] and [[W:Grand stellated 120-cell|grand stellated 120-cell]]. With itself, it can form a [[W:Polytope compound|polytope compound]]: the [[#Symmetries, root systems, and tessellations|compound of two 24-cells]].

=== Related uniform polytopes ===
{{Demitesseract family}}

{{24-cell_family}}

The 24-cell can also be derived as a rectified 16-cell:
{{Tesseract family}}

{{Symmetric_tessellations}}

==See also==
*[[W:Octacube (sculpture)|Octacube (sculpture)]]
*[[W:Uniform 4-polytope#The F4 family|Uniform 4-polytope § The F4 family]]

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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* {{Cite thesis|url= http://resolver.tudelft.nl/uuid:dcffce5a-0b47-404e-8a67-9a3845774d89 |title=Symmetry groups of regular polytopes in three and four dimensions|last=van Ittersum |first=Clara|year=2020|publisher=[[W:Delft University of Technology|Delft University of Technology]]}}
* {{cite arXiv|last1=Kim|first1=Heuna|last2=Rote|first2=G.|date=2016|title=Congruence Testing of Point Sets in 4 Dimensions|class=cs.CG|eprint=1603.07269}}
* {{Cite journal|last1=Perez-Gracia|first1=Alba|last2=Thomas|first2=Federico|date=2017|title=On Cayley's Factorization of 4D Rotations and Applications|url=https://upcommons.upc.edu/bitstream/handle/2117/113067/1749-ON-CAYLEYS-FACTORIZATION-OF-4D-ROTATIONS-AND-APPLICATIONS.pdf|journal=Adv. Appl. Clifford Algebras|volume=27|pages=523–538|doi=10.1007/s00006-016-0683-9|hdl=2117/113067|s2cid=12350382|hdl-access=free}}
* {{Cite journal|last1=Waegell|first1=Mordecai|last2=Aravind|first2=P. K.|date=2009-11-12|title=Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem|journal=Journal of Physics A: Mathematical and Theoretical|volume=43|issue=10|page=105304|language=en|doi=10.1088/1751-8113/43/10/105304|arxiv=0911.2289|s2cid=118501180}}
* {{Cite book|title=Generalized Clifford parallelism|last1=Tyrrell|first1=J. A.|last2=Semple|first2=J.G.|year=1971|publisher=[[W:Cambridge University Press|Cambridge University Press]]|url=https://archive.org/details/generalizedcliff0000tyrr|isbn=0-521-08042-8}}
* {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}}
* {{Cite journal | last1=Mamone|first1=Salvatore | last2=Pileio|first2=Giuseppe | last3=Levitt|first3=Malcolm H. | year=2010 | title=Orientational Sampling Schemes Based on Four Dimensional Polytopes | journal=Symmetry | volume=2 |issue=3 | pages=1423–1449 | doi=10.3390/sym2031423 |bibcode=2010Symm....2.1423M |doi-access=free }}
* {{Cite thesis|title=Applications of Quaternions to Dynamical Simulation, Computer Graphics and Biomechanics|last=Mebius|first=Johan|date=July 2015|publisher=[[W:Delft University of Technology|Delft University of Technology]]|orig-date=11 Jan 1994|doi=10.13140/RG.2.1.3310.3205}}
* {{Cite book|title=Elementary particles and the laws of physics|last1=Feynman|first1=Richard|last2=Weinberg|first2=Steven|publisher=Cambridge University Press|year=1987}}
* {{Cite journal|last=Dorst|first=Leo|title=Conformal Villarceau Rotors|year=2019|journal=Advances in Applied Clifford Algebras|volume=29|issue=44|doi=10.1007/s00006-019-0960-5 |s2cid=253592159 |doi-access=free}}
* {{Cite journal|last1=Koca|first1=Mehmet|last2=Al-Ajmi|first2=Mudhahir|last3=Koc|first3=Ramazan|date=November 2007|title=Polyhedra obtained from Coxeter groups and quaternions|journal=Journal of Mathematical Physics|volume=48|issue=11|pages=113514|doi=10.1063/1.2809467|bibcode=2007JMP....48k3514K |url=https://www.researchgate.net/publication/234907424}}
{{Refend}}

==External links==
* [https://web.archive.org/web/20051118135108/http://valdostamuseum.org/hamsmith/24anime.html 24-cell animations]
* [http://members.home.nl/fg.marcelis/24-cell.htm 24-cell in stereographic projections]
* [http://eusebeia.dyndns.org/4d/24-cell.html 24-cell description and diagrams] {{Webarchive|url=https://web.archive.org/web/20070715053230/http://eusebeia.dyndns.org/4d/24-cell.html |date=2007-07-15 }}
* [https://web.archive.org/web/20071204034724/http://www.xs4all.nl/~jemebius/Ab4help.htm Petrie dodecagons in the 24-cell: mathematics and animation software]

[[Category:Geometry]]
[[Category:Polyscheme]]</text>
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  <page>
    <title>Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 15</title>
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{{Subtitle|Linear subspaces and dual space}}

Linear subspaces of some {{mat|term=K|pm=-}}vector space {{mat|term=V|pm=}} are in direct relation with linear subspaces of the 
{{
Definitionlink
|dual space|
|Context=|
|pm=
}}
{{mat|term= {{op:Dual space|V|}} |pm=.}}

{{:Vector space/Dual space/Linear subspace/Introduction/Section|extra1=&amp;nbsp;In the second semester, when we have inner products at hand, there will also be an orthogonal space for 
{{
Relationchain
| U
|\subseteq| V
|| 
|| 
|| 
|pm=
}}
in {{mat|term=V|pm=}} itself.}}

{{
inputfactproofexercise
|Finite-dimensional vector space/Linear subspace/Kernel/Solution space/Fact|Lemma||
}}

{{Subtitle|The dual mapping}}

{{:Linear mapping/Dual mapping/Introduction/Section}}

{{
inputfactproof
|Linear mapping/To finite-dimensional/Representation with linear forms/Fact|Lemma||extra1=Footnote
}}

{{
inputfactproof
|Dual mapping/Dual basis/Matrix/Fact|Lemma||extra1=Footnote
}}

{{Subtitle|The bidual}}

{{:Vector space/Bidual/Introduction/Section}}

{{List of footnotes}}

}}</text>
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  <page>
    <title>Vector space/Dual space/Linear subspace/Introduction/Section</title>
    <ns>0</ns>
    <id>309948</id>
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Mathematical section{{{opt|}}}
|Content=
{{
inputdefinition
|Linear subspace/Orthogonal space in dual space/Definition||
}}

These orthogonal spaces are linear subspaces of {{mat|term=   {{op:Dual space|V|}} |pm=;}} see
{{
Exerciselink
|Exercisename=
Dual space/Orthogonal space/Linear subspace/Exercise
|Nr=
|pm=.
}}
Whether a linear form {{mat|term= f|pm=}} belongs to {{mathl|term= {{op:Orthogonal space|U}}|pm=}} can be checked on a
{{
Definitionlink
|generating system|
|Context=vs|
|pm=
}}
of {{mat|term= U|pm=;}} see
{{
Exerciselink
|Exercisename=
Dual space/Orthogonal space/Generators/Exercise
|Nr=
|pm=.
}}
The property
{{
Relationchain
| f
|\in|  {{op:Orthogonal space|U|}} 
||
||
||
|pm=
}}
is equivalent with 
{{
Relationchain
| U
| \subseteq |  {{op:Kern|f|}}  
|| 
|| 
|| 
|pm=.
}}
{{{extra1|}}}

{{
inputexample
|Linear subspace/Dual space/1/Example||
}}

{{
inputexample
|Vector space/Basis/Subbasis/Orthogonal space/Example||
}}

{{
inputdefinition
|Linear subspace in dual space/Orthogonal space/Definition||
}}

{{
inputexample
|Linear system/Solution space/Orthogonal space/Example||
}}

In general, we have the relation
{{
Relationchain/display
|  {{op:Orthogonal space|F|}} 
|| \bigcap_{f \in F} {{op:Kern|f|}} 
||
||
||
|pm=.
}}
In particular, 
{{
Relationchain/display
| {{op:Orthogonal space| {{op:Span|f|}} |}} 
|| {{op:Kern|f|}} 
||
||
||
|pm=.
}}

{{
inputfactproof
|Linear subspace/Dual space/Orthogonal space/Correspondence/Fact|Lemma||
}}
|Textform=Section
|Category=
|}}</text>
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  </page>
  <page>
    <title>Linear subspace/Dual space/Orthogonal space/Correspondence/Fact/Proof</title>
    <ns>0</ns>
    <id>309973</id>
    <revision>
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Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
(1) and (2) are clear. (3). The inclusion
{{
Relationchain/display
| U 
|\subseteq | {{op:Orthogonal space|  {{mabr| {{op:Orthogonal space|U|}} |}}  |}} 
||
||
||
|pm=
}}
is also clear. Let
{{
Mathcond|term=
v \in V
||condterm1=
v \notin U
||condterm2=
|pm=.
}}
Then we can choose a 
{{
Definitionlink
|basis|
|Context=vs|
|pm=
}}
{{mathl|term= u_1 {{commadots|}} u_r |pm=}} of {{mat|term= U|pm=}} and extend it to a basis {{mathl|term= u_1 {{commadots|}} u_r, v, v_1 {{commadots|}} v_\ell |pm=}} of {{mat|term= V|pm=.}} The linear form {{mat|term= v^*|pm=}} vanishes on {{mat|term= U|pm=,}} therefore, it belongs to {{mat|term= {{op:Orthogonal space|U|}} |pm=.}} Because of
{{
Relationchain/display
|v^*(v)
|| 1
|\neq|0
||
||
|pm=,
}}
we have
{{
Relationchain
| v
|\notin| {{op:Orthogonal space| {{mabr| {{op:Orthogonal space |U|}} |}} |}}
||
||
||
|pm=.
}}

The inclusion
{{
Relationchain/display
| F
| \subseteq | {{op:Orthogonal space|  {{mabr| {{op:Orthogonal space|F|}} |}}   |}} 
|| 
|| 
|| 
}}
holds immediately. Let
{{
Relationchain
| f
|\in| {{op:Orthogonal space|  {{mabr| {{op:Orthogonal space|F|}} |}}   |}} 
||
||
||
|pm=,
}}
that is, 
{{
Relationchain/display
| {{op:Orthogonal space|F|}}  
|\subseteq|  {{op:Kern|f|}} 
|| 
|| 
|| 
|pm=.
}}
Let {{mathl|term= f_1 {{commadots|}} f_m |}} be a 
{{
Definitionlink
|generating system|
|Context=vs|
|SZ=
}}
of {{mat|term=  F |pm=.}} Due to
{{
Exerciselink
|Exercisename=
Linear forms/Relation between kernels/Linear subspaces/Exercise
}}
we have that {{mat|term= f |}} is a linear combination of the {{mat|term= f_i |pm=;}} therefore, 
{{
Relationchain
| f
|\in| F
||
||
||
|pm=.
}}

(4). We first prove the second part. Let {{mathl|term= f_1 {{commadots|}} f_r|pm=}} be a basis of {{mat|term=  F |pm=,}} and let
{{
Mapping/display
|name=\varphi
|V|K^r
||
|pm=
}}
denote the  mapping where these linear forms are the components. Here, we have
{{
Relationchain/display
|  {{op:Orthogonal space|F|}} 
||  {{op:Kern|\varphi|}} 
||
||
||
|pm=.
}}
Assume that the mapping {{mat|term= \varphi |pm=}} is not surjective. Then {{mathl|term= {{op:Image|\varphi|}} |pm=}} is a strict  linear subspace of {{mat|term= K^r |pm=}} and its dimension is at most {{mathl|term= r-1 |pm=.}} Let {{mat|term= W |pm=}} be a {{mathl|term= (r-1)|pm=-}}dimensional linear subspace with
{{
Relationchain/display
| {{op:Image|\varphi|}}
|\subseteq| W
|\subseteq| K^r
||
||
|pm=.
}}
Due to
{{
Factlink
|Factname=
Hyperplane/Kernel of a linear form/Fact
|Nr=
|pm=,
}}
there is a
{{
Definitionlink
|linear form|
|pm=
}}
{{
Mapping/display
|name= g
|K^r|K
||
|pm=,
}}
whose kernel is exactly {{mat|term= W |pm=.}} Write
{{
Relationchain
|g
|| \sum_{i{{=}} 1}^r a_ip_i
||
||
||
|pm=,
}}
where {{mat|term= p_i |pm=}} is the {{mat|term= i |pm=}}th projection. Then
{{
Relationchain/display
| \sum_{i {{=}} 1}^r a_if_i  
|| g \circ \varphi
|| 0
||
||
|pm=,
}}
contradicting the linear independence of the {{mat|term= f_i|pm=.}} Moreover, {{mat|term= \varphi|pm=}} is surjective and the statement follows from
{{
Factlink
|Factname=
Linear mapping/Dimension formula/Fact
|Nr=
|pm=.
}}

The first part follows by using
{{
Relationchain
| U
|| {{op:Orthogonal space| {{mabr| {{op:Orthogonal space|U|}} |}}  |}} 
|| 
|| 
}}
and applying the second part to
{{
Relationchain
| F
||  {{op:Orthogonal space|U|}}
|| 
|| 
|| 
|pm=.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
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  <page>
    <title>Linear mapping/Dual mapping/Functorial properties/Fact/Proof</title>
    <ns>0</ns>
    <id>310387</id>
    <revision>
      <id>2692157</id>
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      <timestamp>2024-12-16T11:06:49Z</timestamp>
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Mathematical text/Proof
|Text=
{{
Proofstructure
|Strategy=
|Notation=
|Proof=
{{
Enumeration4
|For
{{
Relationchain
| f
|\in| {{op:Dual space|W|}} 
||
||
||
|pm=,
}}
we have
{{
Relationchain/display
|  {{mabr| \varphi \circ \psi |}}^* (f)
|| f \circ {{mabr| \varphi \circ \psi |}} 
|| {{mabr| f \circ \varphi |}} \circ \psi
|| \varphi^*(f) \circ \psi
|| \psi^*( \varphi^*(f) )
|pm=.
}}
|This follows directly from
{{
Relationchain
| f \circ {{op:identity|V|}} 
|| f
||
||
||
|pm=.
}}
|Let
{{
Relationchain
| f
|\in| {{op:Dual space|V|}}
||
||
||
|pm=
}} 
and 
{{
Relationchain/display
|  \psi^*(f) 
|| 0
||
||
||
|pm=.
}}
Because of the surjectivity of {{mat|term= \psi |pm=,}} there exist for every
{{
Relationchain
| v
|\in| V
||
||
||
|pm=
}}
a
{{
Relationchain
| u
|\in| U
||
||
||
|pm=
}}
such that
{{
Relationchain
|\psi (u)
||v
||
||
||
|pm=.
}}
Therefore
{{
Relationchain/display
| f(v)
|| f ( \psi (u))
|| (\psi^*(f)) (u)
|| 0
||
|pm=,
}}
and {{mat|term= f|pm=}} is itself the zero mapping. Due to
{{
Factlink
|Factname=
Linear mapping/Kernel/Injectivity/Fact
|Nr=
|pm=,
}}
{{mat|term= \psi^*|pm=}} injective.
| The condition means that we may consider
{{
Relationchain
| U
|\subseteq| V
|| 
|| 
|| 
|pm=
}} 
as a
{{
Definitionlink
|linear subspace|
|Context=|
|pm=.
}}
Because of
{{
Factlink
|Factname=
Vector space/Finite dimensional/Linear subspace/Direct complement/Fact
|Nr=
|pm=,
}}
we can write
{{
Relationchain/display
|V
|| U \oplus U'
||
||
||
|pm=
}}
with another {{mat|term= K|pm=-}}linear subspace
{{
Relationchain
|U'
| \subseteq | V
||
||
||
|pm=.
}}
A linear form
{{
Mapping/display
|name=g
|U|K
||
|pm=
}}
can always be extended to a linear form
{{
Mapping/display
|name= \tilde{g}
|V|K
||
|pm=,
}}
for example, by defining {{mat|term= \tilde{g} |pm=}} on {{mat|term= U'|pm=}} to be the zero form. This means the surjectivity.
}}
|Closure=
}}
|Textform=Proof 
|Category=See
}}</text>
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    <title>Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 15</title>
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{{Subtitle|Exercise for the break}}

{{
inputexercise
|Fruit seller/Price conditions/Orthogonal space/Exercise||
}}

{{Subtitle|Exercises}}

{{
inputexercise
|R^3/Vector/Orthogonal space/1/Exercise||
}}

{{
inputexercise
|(2,5,-1)/Solution space of a linear system/Exercise||
}}

{{
inputexercise
|Dual space/Orthogonal space/Linear subspace/Exercise||
}}

{{
inputexercise
|Dual space/Orthogonal space/Generators/Exercise||
}}

{{
inputexercise
|Dual space/Orthogonal space/Sum and intersection/Exercise||
}}

{{
inputexercise
|Dual space/Orthogonal space/Intersection and sum/Exercise||
}}

{{
inputexercise
|Dual space/Orthogonal space/Dimension formula/Alternative proof/Exercise||
}}

{{
inputexercise
|Finite-dimensional vector space/Linear subspace/Kernel/Solution space/Fact/Proof/Exercise||
}}

{{
inputexercise
|Symmetric matrices/By linear forms/Exercise||
}}

{{
inputexercise
|Matrices/Multiplication condition/Interpretation/Exercise||
}}

{{
inputexercise
|Matrices/2/U to T/Describe with system of equations/1/Exercise||
}}

{{
inputexercise
|Matrices/3/U to T/Describe with system of equations/1/Exercise||
}}

{{
inputexercise
|Direct sum/Orthogonal spaces in dual space/Exercise||
}}

{{
inputexercise
|Linear mapping/To finite-dimensional/Representation with linear forms/Example/1/Exercise||
}}

{{
inputexercise
|Dual mapping/Rank/Exercise||
}}

{{
inputexercise
|Isomorphism/Dual mapping/Linear subspaces/Exercise||
}}

{{
inputexercise
|Dual mapping/Linear subspaces/Exercise||
}}

{{
inputexercise
|Direct sum/N/Sum/No evaluation/Exercise||
}}

{{
inputexercise
|Vector spaces/Dual mapping/Total assignement/Linearity/Exercise||
}}

{{Subtitle|Hand-in-exercises}}

{{
inputexercise
|Dual space/Linear forms/Linearly independent/Orthogonal space/Exercise|m|
}}

{{
inputexercise
|Matrices/3/U to T/Describe with system of equations/2/Exercise|m|
}}

{{
inputexercise
|Linear mapping/To finite-dimensional/Representation with linear forms/Example/2/Exercise|m|
}}

{{
inputexercise
|Bidual/Finite-dimensional/Orthogonal space/Exercise|m|
}}

}}</text>
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    <title>Linear algebra (Osnabrück 2024-2025)/Finite-dimensional vector space/Linear subspace/Kernel/Solution space/Fact/Proof/Exercise/Exercisereferencenumber</title>
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  <page>
    <title>WikiJournal Preprints/24-cell</title>
    <ns>0</ns>
    <id>313557</id>
    <revision>
      <id>2692093</id>
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      <contributor>
        <username>Dc.samizdat</username>
        <id>2856930</id>
      </contributor>
      <comment>/* Rotations */ cite Goucher 2019 on Spin Groups</comment>
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      <text bytes="231874" sha1="2h201boczku776r6ihf2xth7b8qb15i" xml:space="preserve">{{Article info
|journal=Wikijournal Preprints
|last=Christie
|first=David Brooks
|abstract=The 24-cell is one of only a few uniform polytopes in which the edge length equals the radius. It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. It contains all the convex regular polytopes of four or fewer dimensions made of triangles or squares except the 4-simplex, but it contains no pentagons. It has just four distinct chord lengths, which are the diameters of the hypercubes of dimensions 1 through 4. The 24-cell is the unique construction of these four hypercubic chords and all the regular polytopes that can be built from them. Isoclinic rotations relate the convex regular 4-polytopes to each other, and determine the way they nest inside one another. The 24-cell's characteristic isoclinic rotation takes place in four Clifford parallel great hexagon central planes. It also inherits an isoclinic rotation in six Clifford parallel great square central planes that is characteristic of its three constituent 16-cells. We explore the internal geometry of the 24-cell in detail, as an expression of its rotational symmetries.
|w1=24-cell
}}

== The unique 24-point 24-cell polytope ==

The [[24-cell]] does not have a regular analogue in three dimensions or any other number of dimensions.{{Sfn|Coxeter|1973|p=289|loc=Epilogue|ps=; &quot;Another peculiarity of four-dimensional space is the occurrence of the 24-cell {3,4,3}, which stands quite alone, having no analogue above or below.&quot;}} It is the only one of the six convex regular 4-polytopes which is not the analogue of one of the five Platonic solids. However, it can be seen as the analogue of a pair of irregular solids: the [[W:Cuboctahedron|cuboctahedron]] and its dual the [[W:Rhombic dodecahedron|rhombic dodecahedron]].{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|p=25}}

The 24-cell and the [[W:Tesseract|8-cell (tesseract)]] are the only convex regular 4-polytopes in which the edge length equals the radius. The long radius (center to vertex) of each is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including these two four-dimensional polytopes, the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron. These '''radially equilateral polytopes''' are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.

== The 24-cell in the proper sequence of 4-polytopes ==
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell (4-simplex), those with a 5 in their Schlӓfli symbol,{{Efn|The convex regular polytopes in the first four dimensions with a 5 in their Schlӓfli symbol are the [[W:Pentagon|pentagon]] {5}, the [[W:Icosahedron|icosahedron]] {3, 5}, the [[W:Dodecahedron|dodecahedron]] {5, 3}, the [[600-cell]] {3,3,5} and the [[120-cell]] {5,3,3}. The [[5-cell]] {3, 3, 3} is also pentagonal in the sense that its [[W:Petrie polygon|Petrie polygon]] is the pentagon.|name=pentagonal polytopes|group=}} and the regular polygons with 7 or more sides. In other words, the 24-cell contains ''all'' of the regular polytopes made of triangles and squares that exist in four dimensions except the regular 5-cell, but ''none'' of the pentagonal polytopes. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or [[W:24-cell honeycomb|its honeycomb]].

The 24-cell is the fourth in the sequence of six [[W:Convex regular 4-polytope|convex regular 4-polytope]]s in order of size and complexity. These can be ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. This is their proper order of enumeration: the order in which they nest inside each other as compounds.{{Sfn|Coxeter|1973|loc=§7.8 The enumeration of possible regular figures|p=136}}{{Sfn|Goucher|2020|loc=Subsumptions of regular polytopes}} Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} within the same radius. The 5-cell (4-simplex) is the limit smallest case, and the 120-cell is the largest. Complexity (as measured by comparing [[24-cell#As a configuration|configuration matrices]] or simply the number of vertices) follows the same ordering. This provides an alternative numerical naming scheme for regular polytopes in which the 24-cell is the 24-point 4-polytope: fourth in the ascending sequence that runs from 5-point (5-cell) 4-polytope to 600-point (120-cell) 4-polytope. 

{{Regular convex 4-polytopes|wiki=W:|radius=1}}

The 24-cell can be deconstructed into 3 overlapping instances of its predecessor the [[W:Tesseract|8-cell (tesseract)]], as the 8-cell can be deconstructed into 2 instances of its predecessor the [[16-cell]].{{Sfn|Coxeter|1973|p=302|pp=|loc=Table VI (ii): 𝐈𝐈 = {3,4,3}|ps=: see Result column}} The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length. The edge length will always be different unless predecessor and successor are ''both'' radially equilateral, i.e. their edge length is the same as their radius (so both are preserved). Since radially equilateral polytopes are rare, it seems that the only such construction (in any dimension) is from the 8-cell to the 24-cell, making the 24-cell the unique regular polytope (in any dimension) which has the same edge length as its predecessor of the same radius.

== Coordinates ==
The 24-cell has two natural systems of Cartesian coordinates, which reveal distinct structure.

=== Great squares ===
The 24-cell is the [[W:Convex hull|convex hull]] of its vertices which can be described as the 24 coordinate [[W:Permutation|permutation]]s of:

&lt;math display=&quot;block&quot;&gt;(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .&lt;/math&gt;

Those coordinates{{Sfn|Coxeter|1973|p=156|loc=§8.7. Cartesian Coordinates}} can be constructed as {{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}}, [[W:Rectification (geometry)|rectifying]] the [[16-cell]] {{Coxeter–Dynkin diagram|node_1|3|node|3|node|4|node}} with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the [[W:Octahedron|octahedron]]; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process{{Sfn|Coxeter|1973|p=|pp=145-146|loc=§8.1 The simple truncations of the general regular polytope}} also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.

In this frame of reference the 24-cell has edges of length {{sqrt|2}} and is inscribed in a [[W:3-sphere|3-sphere]] of radius {{sqrt|2}}. Remarkably, the edge length equals the circumradius, as in the [[W:Hexagon|hexagon]], or the [[W:Cuboctahedron|cuboctahedron]].

The 24 vertices form 18 great squares{{Efn|The edges of six of the squares are aligned with the grid lines of the ''{{radic|2}} radius coordinate system''. For example:
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1, −1,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. The edges of the squares are not 24-cell edges, they are interior chords joining two vertices 90&lt;sup&gt;o&lt;/sup&gt; distant from each other; so the squares are merely invisible configurations of four of the 24-cell's vertices, not visible 24-cell features.|name=|group=}} (3 sets of 6 orthogonal{{Efn|Up to 6 planes can be mutually orthogonal in 4 dimensions. 3 dimensional space accommodates only 3 perpendicular axes and 3 perpendicular planes through a single point. In 4 dimensional space we may have 4 perpendicular axes and 6 perpendicular planes through a point (for the same reason that the tetrahedron has 6 edges, not 4): there are 6 ways to take 4 dimensions 2 at a time.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Three such perpendicular planes (pairs of axes) meet at each vertex of the 24-cell (for the same reason that three edges meet at each vertex of the tetrahedron). Each of the 6 planes is [[W:Completely orthogonal|completely orthogonal]] to just one of the other planes: the only one with which it does not share a line (for the same reason that each edge of the tetrahedron is orthogonal to just one of the other edges: the only one with which it does not share a point). Two completely orthogonal planes are perpendicular and opposite each other, as two edges of the tetrahedron are perpendicular and opposite.|name=six orthogonal planes tetrahedral symmetry}} central squares), 3 of which intersect at each vertex. By viewing just one square at each vertex, the 24-cell can be seen as the vertices of 3 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} great squares which intersect{{Efn|Two planes in 4-dimensional space can have four possible reciprocal positions: (1) they can coincide (be exactly the same plane); (2) they can be parallel (the only way they can fail to intersect at all); (3) they can intersect in a single line, as two non-parallel planes do in 3-dimensional space; or (4) '''they can intersect in a single point'''{{Efn|To visualize how two planes can intersect in a single point in a four dimensional space, consider the Euclidean space (w, x, y, z) and imagine that the w dimension represents time rather than a spatial dimension. The xy central plane (where w{{=}}0, z{{=}}0) shares no axis with the wz central plane (where x{{=}}0, y{{=}}0). The xy plane exists at only a single instant in time (w{{=}}0); the wz plane (and in particular the w axis) exists all the time. Thus their only moment and place of intersection is at the origin point (0,0,0,0).|name=how planes intersect at a single point}} if they are [[W:Completely orthogonal|completely orthogonal]].|name=how planes intersect}} at no vertices.{{Efn|name=three square fibrations}}

=== Great hexagons ===
The 24-cell is [[W:Self-dual|self-dual]], having the same number of vertices (24) as cells and the same number of edges (96) as faces.

If the dual of the above 24-cell of edge length {{sqrt|2}} is taken by reciprocating it about its ''inscribed'' sphere, another 24-cell is found which has edge length and circumradius 1, and its coordinates reveal more structure. In this frame of reference the 24-cell lies vertex-up, and its vertices can be given as follows:

8 vertices obtained by permuting the ''integer'' coordinates:

&lt;math display=&quot;block&quot;&gt;\left( \pm 1, 0, 0, 0 \right)&lt;/math&gt;

and 16 vertices with ''half-integer'' coordinates of the form:

&lt;math display=&quot;block&quot;&gt;\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)&lt;/math&gt;

all 24 of which lie at distance 1 from the origin.

[[24-cell#Quaternionic interpretation|Viewed as quaternions]],{{Efn|In [[W:Euclidean geometry#Higher dimensions|four-dimensional Euclidean geometry]], a [[W:Quaternion|quaternion]] is simply a (w, x, y, z) Cartesian coordinate. [[W:William Rowan Hamilton|Hamilton]] did not see them as such when he [[W:History of quaternions|discovered the quaternions]]. [[W:Ludwig Schläfli|Schläfli]] would be the first to consider [[W:4-dimensional space|four-dimensional Euclidean space]], publishing his discovery of the regular [[W:Polyscheme|polyscheme]]s in 1852, but Hamilton would never be influenced by that work, which remained obscure into the 20th century. Hamilton found the quaternions when he realized that a fourth dimension, in some sense, would be necessary in order to model rotations in three-dimensional space.{{Sfn|Stillwell|2001|p=18-21}} Although he described a quaternion as an ''ordered four-element multiple of real numbers'', the quaternions were for him an extension of the complex numbers, not a Euclidean space of four dimensions.|name=quaternions}} these are the unit [[W:Hurwitz quaternions|Hurwitz quaternions]]. These 24 quaternions represent (in antipodal pairs) the 12 rotations of a regular tetrahedron.{{Sfn|Stillwell|2001|p=22}}

The 24-cell has unit radius and unit edge length in this coordinate system. We refer to the system as ''unit radius coordinates'' to distinguish it from others, such as the {{sqrt|2}} radius coordinates used to reveal the great [[#Great squares|squares]] above.{{Efn|The edges of the orthogonal great squares are ''not'' aligned with the grid lines of the ''unit radius coordinate system''. Six of the squares do lie in the 6 orthogonal planes of this coordinate system, but their edges are the {{sqrt|2}} ''diagonals'' of unit edge length squares of the coordinate lattice. For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}0, −1,{{spaces|2}}0,{{spaces|2}}0){{spaces|3}}({{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0,{{spaces|2}}0)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is the square in the ''xy'' plane. Notice that the 8 ''integer'' coordinates comprise the vertices of the 6 orthogonal squares.|name=orthogonal squares|group=}}

The 24 vertices and 96 edges form 16 non-orthogonal great hexagons,{{Efn|The hexagons are inclined (tilted) at 60 degrees with respect to the unit radius coordinate system's orthogonal planes. Each hexagonal plane contains only ''one'' of the 4 coordinate system axes.{{Efn|Each great hexagon of the 24-cell contains one axis (one pair of antipodal vertices) belonging to each of the three inscribed 16-cells. The 24-cell contains three disjoint inscribed 16-cells, rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other (so their corresponding vertices are 120° {{=}} {{radic|3}} apart). A [[16-cell#Coordinates|16-cell is an orthonormal ''basis'']] for a 4-dimensional coordinate system, because its 8 vertices define the four orthogonal axes. In any choice of a vertex-up coordinate system (such as the unit radius coordinates used in this article), one of the three inscribed 16-cells is the basis for the coordinate system, and each hexagon has only ''one'' axis which is a coordinate system axis.|name=three basis 16-cells}} The hexagon consists of 3 pairs of opposite vertices (three 24-cell diameters): one opposite pair of ''integer'' coordinate vertices (one of the four coordinate axes), and two opposite pairs of ''half-integer'' coordinate vertices (not coordinate axes). For example:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
is a hexagon on the ''y'' axis. Unlike the {{sqrt|2}} squares, the hexagons are actually made of 24-cell edges, so they are visible features of the 24-cell.|name=non-orthogonal hexagons|group=}} four of which intersect{{Efn||name=how planes intersect}} at each vertex.{{Efn|It is not difficult to visualize four hexagonal planes intersecting at 60 degrees to each other, even in three dimensions. Four hexagonal central planes intersect at 60 degrees in the [[W:Cuboctahedron|cuboctahedron]]. Four of the 24-cell's 16 hexagonal central planes (lying in the same 3-dimensional hyperplane) intersect at each of the 24-cell's vertices exactly the way they do at the center of a cuboctahedron. But the ''edges'' around the vertex do not meet as the radii do at the center of a cuboctahedron; the 24-cell has 8 edges around each vertex, not 12, so its vertex figure is the cube, not the cuboctahedron. The 8 edges meet exactly the way 8 edges do at the apex of a canonical [[W:Cubic pyramid|cubic pyramid]].{{Efn|name=24-cell vertex figure}}|name=cuboctahedral hexagons}} By viewing just one hexagon at each vertex, the 24-cell can be seen as the 24 vertices of 4 non-intersecting hexagonal great circles which are [[W:Clifford parallel|Clifford parallel]] to each other.{{Efn|name=four hexagonal fibrations}}

The 12 axes and 16 hexagons of the 24-cell constitute a [[W:Reye configuration|Reye configuration]], which in the language of [[W:Configuration (geometry)|configurations]] is written as 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt; to indicate that each axis belongs to 4 hexagons, and each hexagon contains 3 axes.{{Sfn|Waegell|Aravind|2009|loc=§3.4 The 24-cell: points, lines and Reye's configuration|pp=4-5|ps=; In the 24-cell Reye's &quot;points&quot; and &quot;lines&quot; are axes and hexagons, respectively.}}

=== Great triangles ===
The 24 vertices form 32 equilateral great triangles, of edge length {{radic|3}} in the unit-radius 24-cell,{{Efn|These triangles' edges of length {{sqrt|3}} are the diagonals{{Efn|name=missing the nearest vertices}} of cubical cells of unit edge length found within the 24-cell, but those cubical (tesseract){{Efn|name=three 8-cells}} cells are not cells of the unit radius coordinate lattice.|name=cube diagonals}} inscribed in the 16 great hexagons.{{Efn|These triangles lie in the same planes containing the hexagons;{{Efn|name=non-orthogonal hexagons}} two triangles of edge length {{sqrt|3}} are inscribed in each hexagon.  For example, in unit radius coordinates:
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0,{{spaces|2}}1,{{spaces|2}}0)
{{indent|5}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}({{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|5}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;){{spaces|3}}(−&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;, −&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;,{{spaces|2}}&lt;small&gt;{{sfrac|1|2}}&lt;/small&gt;)
{{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)&lt;br&gt;
are two opposing central triangles on the ''y'' axis, with each triangle formed by the vertices in alternating rows. Unlike the hexagons, the {{sqrt|3}} triangles are not made of actual 24-cell edges, so they are invisible features of the 24-cell, like the {{sqrt|2}} squares.|name=central triangles|group=}} 

Each great triangle is a ring linking three completely disjoint{{Efn|name=completely disjoint}} great squares. The 18 great squares of the 24-cell occur as three sets of 6 orthogonal great squares,{{Efn|name=Six orthogonal planes of the Cartesian basis}} each forming a [[16-cell]].{{Efn|name=three isoclinic 16-cells}} The three 16-cells are completely disjoint (and [[#Clifford parallel polytopes|Clifford parallel]]): each has its own 8 vertices (on 4 orthogonal axes) and its own 24 edges (of length {{radic|2}}). The 18 square great circles are crossed by 16 hexagonal great circles; each hexagon has one axis (2 vertices) in each 16-cell.{{Efn|name=non-orthogonal hexagons}} The two great triangles inscribed in each great hexagon (occupying its alternate vertices, and with edges that are its {{radic|3}} chords) have one vertex in each 16-cell. Thus ''each great triangle is a ring linking the three completely disjoint 16-cells''. There are four different ways (four different ''fibrations'' of the 24-cell) in which the 8 vertices of the 16-cells correspond by being triangles of vertices {{radic|3}} apart: there are 32 distinct linking triangles. Each ''pair'' of 16-cells forms an 8-cell (tesseract).{{Efn|name=three 16-cells form three tesseracts}} Each great triangle has one {{radic|3}} edge in each tesseract, so it is also a ring linking the three tesseracts.

== Hypercubic chords ==
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-Dimensional Polytopes in Parallel Solid Sections (§13.1); (i) Sections of {3,4,3} (edge 2) beginning with a vertex; see column ''a''|5=}} at four different [[W:Chord (geometry)|chord]] lengths from each other: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}. The {{sqrt|1}} chords (the 24-cell edges) are the edges of central hexagons, and the {{sqrt|3}} chords are the diagonals of central hexagons. The {{sqrt|2}} chords are the edges of central squares, and the {{sqrt|4}} chords are the diagonals of central squares.

Each vertex is joined to 8 others{{Efn|The 8 nearest neighbor vertices surround the vertex (in the curved 3-dimensional space of the 24-cell's boundary surface) the way a cube's 8 corners surround its center. (The [[W:Vertex figure|vertex figure]] of the 24-cell is a cube.)|name=8 nearest vertices}} by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices{{Efn|The 6 second-nearest neighbor vertices surround the vertex in curved 3-dimensional space the way an octahedron's 6 corners surround its center.|name=6 second-nearest vertices}} located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

To visualize how the interior polytopes of the 24-cell fit together (as described [[#Constructions|below]]), keep in mind that the four chord lengths ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the long diameters of the [[W:Hypercube|hypercube]]s of dimensions 1 through 4: the long diameter of the square is {{sqrt|2}}; the long diameter of the cube is {{sqrt|3}}; and the long diameter of the tesseract is {{sqrt|4}}.{{Efn|Thus ({{sqrt|1}}, {{sqrt|2}}, {{sqrt|3}}, {{sqrt|4}}) are the vertex chord lengths of the tesseract as well as of the 24-cell. They are also the diameters of the tesseract (from short to long), though not of the 24-cell.}} Moreover, the long diameter of the octahedron is {{sqrt|2}} like the square; and the long diameter of the 24-cell itself is {{sqrt|4}} like the tesseract.

== Geodesics ==
The vertex chords of the 24-cell are arranged in [[W:Geodesic|geodesic]] [[W:great circle|great circle]] polygons.{{Efn|A geodesic great circle lies in a 2-dimensional plane which passes through the center of the polytope. Notice that in 4 dimensions this central plane does ''not'' bisect the polytope into two equal-sized parts, as it would in 3 dimensions, just as a diameter (a central line) bisects a circle but does not bisect a sphere. Another difference is that in 4 dimensions not all pairs of great circles intersect at two points, as they do in 3 dimensions; some pairs do, but some pairs of great circles are non-intersecting Clifford parallels.{{Efn|name=Clifford parallels}}}} The [[W:Geodesic distance|geodesic distance]] between two 24-cell vertices along a path of {{sqrt|1}} edges is always 1, 2, or 3, and it is 3 only for opposite vertices.{{Efn|If the [[W:Euclidean distance|Pythagorean distance]] between any two vertices is {{sqrt|1}}, their geodesic distance is 1; they may be two adjacent vertices (in the curved 3-space of the surface), or a vertex and the center (in 4-space). If their Pythagorean distance is {{sqrt|2}}, their geodesic distance is 2 (whether via 3-space or 4-space, because the path along the edges is the same straight line with one 90&lt;sup&gt;o&lt;/sup&gt; bend in it as the path through the center). If their Pythagorean distance is {{sqrt|3}}, their geodesic distance is still 2 (whether on a hexagonal great circle past one 60&lt;sup&gt;o&lt;/sup&gt; bend, or as a straight line with one 60&lt;sup&gt;o&lt;/sup&gt; bend in it through the center). Finally, if their Pythagorean distance is {{sqrt|4}}, their geodesic distance is still 2 in 4-space (straight through the center), but it reaches 3 in 3-space (by going halfway around a hexagonal great circle).|name=Geodesic distance}}

The {{sqrt|1}} edges occur in 16 [[#Great hexagons|hexagonal great circles]] (in planes inclined at 60 degrees to each other), 4 of which cross{{Efn|name=cuboctahedral hexagons}} at each vertex.{{Efn|Eight {{sqrt|1}} edges converge in curved 3-dimensional space from the corners of the 24-cell's cubical vertex figure{{Efn|The [[W:Vertex figure|vertex figure]] is the facet which is made by truncating a vertex; canonically, at the mid-edges incident to the vertex. But one can make similar vertex figures of different radii by truncating at any point along those edges, up to and including truncating at the adjacent vertices to make a ''full size'' vertex figure. Stillwell defines the vertex figure as &quot;the convex hull of the neighbouring vertices of a given vertex&quot;.{{Sfn|Stillwell|2001|p=17}} That is what serves the illustrative purpose here.|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell. The straight lines are geodesics: two {{sqrt|1}}-length segments of an apparently straight line (in the 3-space of the 24-cell's curved surface) that is bent in the 4th dimension into a great circle hexagon (in 4-space). Imagined from inside this curved 3-space, the bends in the hexagons are invisible. From outside (if we could view the 24-cell in 4-space), the straight lines would be seen to bend in the 4th dimension at the cube centers, because the center is displaced outward in the 4th dimension, out of the hyperplane defined by the cube's vertices. Thus the vertex cube is actually a [[W:Cubic pyramid|cubic pyramid]]. Unlike a cube, it seems to be radially equilateral (like the tesseract and the 24-cell itself): its &quot;radius&quot; equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space &lt;math&gt;\mathbb{R}^3&lt;/math&gt;, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In 4-space the 8 edges radiating from its apex are not actually its radii: the apex of the [[W:Cubic pyramid|cubic pyramid]] is not actually its center, just one of its vertices. But in curved 3-space the edges radiating symmetrically from the apex ''are'' radii, so the cube is radially equilateral ''in that curved 3-space'' &lt;math&gt;\mathbb{S}^3&lt;/math&gt;. In Euclidean 4-space &lt;math&gt;\mathbb{R}^4&lt;/math&gt; 24 edges radiating symmetrically from a central point make the radially equilateral 24-cell, and a symmetrical subset of 16 of those edges make the [[W:Tesseract#Radial equilateral symmetry|radially equilateral tesseract]].}}|name=24-cell vertex figure}} The 96 distinct {{sqrt|1}} edges divide the surface into 96 triangular faces and 24 octahedral cells: a 24-cell. The 16 hexagonal great circles can be divided into 4 sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 4 hexagons in each set reach all 24 vertices.{{Efn|name=hexagonal fibrations}}

The {{sqrt|2}} chords occur in 18 [[#Great squares|square great circles]] (3 sets of 6 orthogonal planes{{Efn|name=Six orthogonal planes of the Cartesian basis}}), 3 of which cross at each vertex.{{Efn|Six {{sqrt|2}} chords converge in 3-space from the face centers of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 3 straight lines which cross there perpendicularly. The 8 vertices of the cube are the eight nearest other vertices of the 24-cell, and eight {{sqrt|1}} edges converge from there, but let us ignore them now, since 7 straight lines crossing at the center is confusing to visualize all at once. Each of the six {{sqrt|2}} chords runs from this cube's center (the vertex) through a face center to the center of an adjacent (face-bonded) cube, which is another vertex of the 24-cell: not a nearest vertex (at the cube corners), but one located 90° away in a second concentric shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices. The face-center through which the {{sqrt|2}} chord passes is the mid-point of the {{sqrt|2}} chord, so it lies inside the 24-cell.|name=|group=}} The 72 distinct {{sqrt|2}} chords do not run in the same planes as the hexagonal great circles; they do not follow the 24-cell's edges, they pass through its octagonal cell centers.{{Efn|One can cut the 24-cell through 6 vertices (in any hexagonal great circle plane), or through 4 vertices (in any square great circle plane). One can see this in the [[W:Cuboctahedron|cuboctahedron]] (the central [[W:hyperplane|hyperplane]] of the 24-cell), where there are four hexagonal great circles (along the edges) and six square great circles (across the square faces diagonally).}} The 72 {{sqrt|2}} chords are the 3 orthogonal axes of the 24 octahedral cells, joining vertices which are 2 {{radic|1}} edges apart. The 18 square great circles can be divided into 3 sets of 6 non-intersecting Clifford parallel geodesics,{{Efn|[[File:Hopf band wikipedia.png|thumb|Two [[W:Clifford parallel|Clifford parallel]] [[W:Great circle|great circle]]s on the [[W:3-sphere|3-sphere]] spanned by a twisted [[W:Annulus (mathematics)|annulus]]. They have a common center point in [[W:Rotations in 4-dimensional Euclidean space|4-dimensional Euclidean space]], and could lie in [[W:Completely orthogonal|completely orthogonal]] rotation planes.]][[W:Clifford parallel|Clifford parallel]]s are non-intersecting curved lines that are parallel in the sense that the perpendicular (shortest) distance between them is the same at each point.{{Sfn|Tyrrell|Semple|1971|loc=§3. Clifford's original definition of parallelism|pp=5-6}} A double helix is an example of Clifford parallelism in ordinary 3-dimensional Euclidean space. In 4-space Clifford parallels occur as geodesic great circles on the [[W:3-sphere|3-sphere]].{{Sfn|Kim|Rote|2016|pp=8-10|loc=Relations to Clifford Parallelism}} Whereas in 3-dimensional space, any two geodesic great circles on the 2-sphere will always intersect at two antipodal points, in 4-dimensional space not all great circles intersect; various sets of Clifford parallel non-intersecting geodesic great circles can be found on the 3-sphere. Perhaps the simplest example is that six mutually orthogonal great circles can be drawn on the 3-sphere, as three pairs of completely orthogonal great circles.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each completely orthogonal pair is Clifford parallel. The two circles cannot intersect at all, because they lie in planes which intersect at only one point: the center of the 3-sphere.{{Efn|Each square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal).|name=only some Clifford parallels are orthogonal}} Because they are perpendicular and share a common center,{{Efn|In 4-space, two great circles can be perpendicular and share a common center ''which is their only point of intersection'', because there is more than one great [[W:2-sphere|2-sphere]] on the [[W:3-sphere|3-sphere]]. The dimensionally analogous structure to a [[W:Great circle|great circle]] (a great 1-sphere) is a great 2-sphere,{{Sfn|Stillwell|2001|p=24}} which is an ordinary sphere that constitutes an ''equator'' boundary dividing the 3-sphere into two equal halves, just as a great circle divides the 2-sphere. Although two Clifford parallel great circles{{Efn|name=Clifford parallels}} occupy the same 3-sphere, they lie on different great 2-spheres. The great 2-spheres are [[#Clifford parallel polytopes|Clifford parallel 3-dimensional objects]], displaced relative to each other by a fixed distance ''d'' in the fourth dimension. Their corresponding points (on their two surfaces) are ''d'' apart. The 2-spheres (by which we mean their surfaces) do not intersect at all, although they have a common center point in 4-space. The displacement ''d'' between a pair of their corresponding points is the [[#Geodesics|chord of a great circle]] which intersects both 2-spheres, so ''d'' can be represented equivalently as a linear chordal distance, or as an angular distance.|name=great 2-spheres}} the two circles are obviously not parallel and separate in the usual way of parallel circles in 3 dimensions; rather they are connected like adjacent links in a chain, each passing through the other without intersecting at any points, forming a [[W:Hopf link|Hopf link]].|name=Clifford parallels}} such that only one square great circle in each set passes through each vertex, and the 6 squares in each set reach all 24 vertices.{{Efn|name=square fibrations}}

The {{sqrt|3}} chords occur in 32 [[#Great triangles|triangular great circles]] in 16 planes, 4 of which cross at each vertex.{{Efn|Eight {{sqrt|3}} chords converge from the corners of the 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} and meet at its center (the vertex), where they form 4 straight lines which cross there. Each of the eight {{sqrt|3}} chords runs from this cube's center to the center of a diagonally adjacent (vertex-bonded) cube,{{Efn|name=missing the nearest vertices}} which is another vertex of the 24-cell: one located 120° away in a third concentric shell of eight {{sqrt|3}}-distant vertices surrounding the second shell of six {{sqrt|2}}-distant vertices that surrounds the first shell of eight {{sqrt|1}}-distant vertices.|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} The 96 distinct {{sqrt|3}} chords{{Efn|name=cube diagonals}} run vertex-to-every-other-vertex in the same planes as the hexagonal great circles.{{Efn|name=central triangles}} They are the 3 edges of the 32 great triangles inscribed in the 16 great hexagons, joining vertices which are 2 {{sqrt|1}} edges apart on a great circle.{{Efn|name=three 8-cells}}

The {{sqrt|4}} chords occur as 12 vertex-to-vertex diameters (3 sets of 4 orthogonal axes), the 24 radii around the 25th central vertex.

The sum of the squared lengths{{Efn|The sum of 1・96 + 2・72 + 3・96 + 4・12 is 576.}} of all these distinct chords of the 24-cell is 576 = 24&lt;sup&gt;2&lt;/sup&gt;.{{Efn|The sum of the squared lengths of all the distinct chords of any regular convex n-polytope of unit radius is the square of the number of vertices.{{Sfn|Copher|2019|loc=§3.2 Theorem 3.4|p=6}}}} These are all the central polygons through vertices, but in 4-space there are geodesics on the 3-sphere which do not lie in central planes at all. There are geodesic shortest paths between two 24-cell vertices that are helical rather than simply circular; they correspond to diagonal [[#Isoclinic rotations|isoclinic rotations]] rather than [[#Simple rotations|simple rotations]].{{Efn|name=isoclinic geodesic}}

The {{sqrt|1}} edges occur in 48 parallel pairs, {{sqrt|3}} apart. The {{sqrt|2}} chords occur in 36 parallel pairs, {{sqrt|2}} apart. The {{sqrt|3}} chords occur in 48 parallel pairs, {{sqrt|1}} apart.{{Efn|Each pair of parallel {{sqrt|1}} edges joins a pair of parallel {{sqrt|3}} chords to form one of 48 rectangles (inscribed in the 16 central hexagons), and each pair of parallel {{sqrt|2}} chords joins another pair of parallel {{sqrt|2}} chords to form one of the 18 central squares.|name=|group=}}

The central planes of the 24-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming a [[W:Cuboctahedron|cuboctahedron]]. The great hexagons are 60 degrees apart; the great squares are 90 degrees or 60 degrees apart; a great square and a great hexagon are 90 degrees ''and'' 60 degrees apart.{{Efn|Two angles are required to fix the relative positions of two planes in 4-space.{{Sfn|Kim|Rote|2016|p=7|loc=§6 Angles between two Planes in 4-Space|ps=; &quot;In four (and higher) dimensions, we need two angles to fix the relative position between two planes. (More generally, ''k'' angles are defined between ''k''-dimensional subspaces.)&quot;.}} Since all planes in the same hyperplane{{Efn|One way to visualize the ''n''-dimensional [[W:Hyperplane|hyperplane]]s is as the ''n''-spaces which can be defined by ''n + 1'' points. A point is the 0-space which is defined by 1 point. A line is the 1-space which is defined by 2 points which are not coincident. A plane is the 2-space which is defined by 3 points which are not colinear (any triangle). In 4-space, a 3-dimensional hyperplane is the 3-space which is defined by 4 points which are not coplanar (any tetrahedron). In 5-space, a 4-dimensional hyperplane is the 4-space which is defined by 5 points which are not cocellular (any 5-cell). These [[W:Simplex|simplex]] figures divide the hyperplane into two parts (inside and outside the figure), but in addition they divide the enclosing space into two parts (above and below the hyperplane). The ''n'' points ''bound'' a finite simplex figure (from the outside), and they ''define'' an infinite hyperplane (from the inside).{{Sfn|Coxeter|1973|loc=§7.2.|p=120|ps=: &quot;... any ''n''+1 points which do not lie in an (''n''-1)-space are the vertices of an ''n''-dimensional ''simplex''.... Thus the general simplex may alternatively be defined as a finite region of ''n''-space enclosed by ''n''+1 ''hyperplanes'' or (''n''-1)-spaces.&quot;}} These two divisions are orthogonal, so the defining simplex divides space into six regions: inside the simplex and in the hyperplane, inside the simplex but above or below the hyperplane, outside the simplex but in the hyperplane, and outside the simplex above or below the hyperplane.|name=hyperplanes|group=}} are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great hexagons in different hyperplanes are 60 degrees apart in ''both'' angles. Great squares in different hyperplanes are 90 degrees apart in ''both'' angles ([[W:Completely orthogonal|completely orthogonal]]) or 60 degrees apart in ''both'' angles.{{Efn||name=Clifford parallel squares in the 16-cell and 24-cell}} Planes which are separated by two equal angles are called ''isoclinic''. Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great square and a great hexagon in different hyperplanes ''may'' be isoclinic, but often they are separated by a 90 degree angle ''and'' a 60 degree angle.|name=two angles between central planes}} Each set of similar central polygons (squares or hexagons) can be divided into 4 sets of non-intersecting Clifford parallel polygons (of 6 squares or 4 hexagons).{{Efn|Each pair of Clifford parallel polygons lies in two different hyperplanes (cuboctahedrons). The 4 Clifford parallel hexagons lie in 4 different cuboctahedrons.}} Each set of Clifford parallel great circles is a parallel [[W:Hopf fibration|fiber bundle]] which visits all 24 vertices just once.

Each great circle intersects{{Efn|name=how planes intersect}} with the other great circles to which it is not Clifford parallel at one {{sqrt|4}} diameter of the 24-cell.{{Efn|Two intersecting great squares or great hexagons share two opposing vertices, but squares or hexagons on Clifford parallel great circles share no vertices. Two intersecting great triangles share only one vertex, since they lack opposing vertices.|name=how great circle planes intersect|group=}} Great circles which are [[W:Completely orthogonal|completely orthogonal]] or otherwise Clifford parallel{{Efn|name=Clifford parallels}} do not intersect at all: they pass through disjoint sets of vertices.{{Efn|name=pairs of completely orthogonal planes}}

== Constructions ==
Triangles and squares come together uniquely in the 24-cell to generate, as interior features,{{Efn|Interior features are not considered elements of the polytope. For example, the center of a 24-cell is a noteworthy feature (as are its long radii), but these interior features do not count as elements in [[#Configuration|its configuration matrix]], which counts only elementary features (which are not interior to any other feature including the polytope itself). Interior features are not rendered in most of the diagrams and illustrations in this article (they are normally invisible). In illustrations showing interior features, we always draw interior edges as dashed lines, to distinguish them from elementary edges.|name=interior features|group=}} all of the triangle-faced and square-faced regular convex polytopes in the first four dimensions (with caveats for the [[5-cell]] and the [[600-cell]]).{{Efn|The [[600-cell]] is larger than the 24-cell, and contains the 24-cell as an interior feature.{{Sfn|Coxeter|1973|p=153|loc=8.5. Gosset's construction for {3,3,5}|ps=: &quot;In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s.&quot;}} The regular [[5-cell]] is not found in the interior of any convex regular 4-polytope except the [[120-cell]],{{Sfn|Coxeter|1973|p=304|loc=Table VI(iv) II={5,3,3}|ps=: Faceting {5,3,3}[120𝛼&lt;sub&gt;4&lt;/sub&gt;]{3,3,5} of the 120-cell reveals 120 regular 5-cells.}} though every convex 4-polytope can be [[#Characteristic orthoscheme|deconstructed into irregular 5-cells.]]|name=|group=}} Consequently, there are numerous ways to construct or deconstruct the 24-cell.

==== Reciprocal constructions from 8-cell and 16-cell ====
The 8 integer vertices (±1, 0, 0, 0) are the vertices of a regular [[16-cell]], and the 16 half-integer vertices (±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) are the vertices of its dual, the [[W:Tesseract|8-cell (tesseract)]].{{Sfn|Egan|2021|loc=animation of a rotating 24-cell|ps=: {{color|red}} half-integer vertices (tesseract), {{Font color|fg=yellow|bg=black|text=yellow}} and {{color|black}} integer vertices (16-cell).}} The tesseract gives Gosset's construction{{Sfn|Coxeter|1973|p=150|loc=Gosset}} of the 24-cell, equivalent to cutting a tesseract into 8 [[W:Cubic pyramid|cubic pyramid]]s, and then attaching them to the facets of a second tesseract. The analogous construction in 3-space gives the [[W:Rhombic dodecahedron|rhombic dodecahedron]] which, however, is not regular.{{Efn|[[File:R1-cube.gif|thumb|150px|Construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube.]]This animation shows the construction of a [[W:Rhombic dodecahedron|rhombic dodecahedron]] from a cube, by inverting the center-to-face pyramids of a cube. Gosset's construction of a 24-cell from a tesseract is the 4-dimensional analogue of this process, inverting the center-to-cell pyramids of an 8-cell (tesseract).{{Sfn|Coxeter|1973|p=150|loc=Gosset}}|name=rhombic dodecahedron from a cube}} The 16-cell gives the reciprocal construction of the 24-cell, Cesaro's construction,{{Sfn|Coxeter|1973|p=148|loc=§8.2. Cesaro's construction for {3, 4, 3}.}} equivalent to rectifying a 16-cell (truncating its corners at the mid-edges, as described [[#Great squares|above]]). The analogous construction in 3-space gives the [[W:Cuboctahedron|cuboctahedron]] (dual of the rhombic dodecahedron) which, however, is not regular. The tesseract and the 16-cell are the only regular 4-polytopes in the 24-cell.{{Sfn|Coxeter|1973|p=302|loc=Table VI(ii) II={3,4,3}, Result column}}

We can further divide the 16 half-integer vertices into two groups: those whose coordinates contain an even number of minus (−) signs and those with an odd number. Each of these groups of 8 vertices also define a regular 16-cell. This shows that the vertices of the 24-cell can be grouped into three disjoint sets of eight with each set defining a regular 16-cell, and with the complement defining the dual tesseract.{{Sfn|Coxeter|1973|pp=149-150|loc=§8.22. see illustrations Fig. 8.2&lt;small&gt;A&lt;/small&gt; and Fig 8.2&lt;small&gt;B&lt;/small&gt;|p=|ps=}} This also shows that the symmetries of the 16-cell form a subgroup of index 3 of the symmetry group of the 24-cell.{{Efn|name=three 16-cells form three tesseracts}}

==== Diminishings ====
We can [[W:Faceting|facet]] the 24-cell by cutting{{Efn|We can cut a vertex off a polygon with a 0-dimensional cutting instrument (like the point of a knife, or the head of a zipper) by sweeping it along a 1-dimensional line, exposing a new edge. We can cut a vertex off a polyhedron with a 1-dimensional cutting edge (like a knife) by sweeping it through a 2-dimensional face plane, exposing a new face. We can cut a vertex off a polychoron (a 4-polytope) with a 2-dimensional cutting plane (like a snowplow), by sweeping it through a 3-dimensional cell volume, exposing a new cell. Notice that as within the new edge length of the polygon or the new face area of the polyhedron, every point within the new cell volume is now exposed on the surface of the polychoron.}} through interior cells bounded by vertex chords to remove vertices, exposing the [[W:Facet (geometry)|facets]] of interior 4-polytopes [[W:Inscribed figure|inscribed]] in the 24-cell. One can cut a 24-cell through any planar hexagon of 6 vertices, any planar rectangle of 4 vertices, or any triangle of 3 vertices. The great circle central planes ([[#Geodesics|above]]) are only some of those planes. Here we shall expose some of the others: the face planes{{Efn|Each cell face plane intersects with the other face planes of its kind to which it is not completely orthogonal or parallel at their characteristic vertex chord edge. Adjacent face planes of orthogonally-faced cells (such as cubes) intersect at an edge since they are not completely orthogonal.{{Efn|name=how planes intersect}} Although their dihedral angle is 90 degrees in the boundary 3-space, they lie in the same hyperplane{{Efn|name=hyperplanes}} (they are coincident rather than perpendicular in the fourth dimension); thus they intersect in a line, as non-parallel planes do in any 3-space.|name=how face planes intersect}} of interior polytopes.{{Efn|The only planes through exactly 6 vertices of the 24-cell (not counting the central vertex) are the '''16 hexagonal great circles'''. There are no planes through exactly 5 vertices. There are several kinds of planes through exactly 4 vertices: the 18 {{sqrt|2}} square great circles, the '''72 {{sqrt|1}} square (tesseract) faces''', and 144 {{sqrt|1}} by {{sqrt|2}} rectangles. The planes through exactly 3 vertices are the 96 {{sqrt|2}} equilateral triangle (16-cell) faces, and the '''96 {{sqrt|1}} equilateral triangle (24-cell) faces'''. There are an infinite number of central planes through exactly two vertices (great circle [[W:Digon|digon]]s); 16 are distinguished, as each is [[W:Completely orthogonal|completely orthogonal]] to one of the 16 hexagonal great circles. '''Only the polygons composed of 24-cell {{radic|1}} edges are visible''' in the projections and rotating animations illustrating this article; the others contain invisible interior chords.{{Efn|name=interior features}}|name=planes through vertices|group=}}

===== 8-cell =====
Starting with a complete 24-cell, remove 8 orthogonal vertices (4 opposite pairs on 4 perpendicular axes), and the 8 edges which radiate from each, by cutting through 8 cubic cells bounded by {{sqrt|1}} edges to remove 8 [[W:Cubic pyramid|cubic pyramid]]s whose [[W:Apex (geometry)|apexes]] are the vertices to be removed. This removes 4 edges from each hexagonal great circle (retaining just one opposite pair of edges), so no continuous hexagonal great circles remain. Now 3 perpendicular edges meet and form the corner of a cube at each of the 16 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to a tetrahedral vertex figure (see [[#Relationships among interior polytopes|Kepler's drawing]]). The vertex cube has vanished, and now there are only 4 corners of the vertex figure where before there were 8. Four tesseract edges converge from the tetrahedron vertices and meet at its center, where they do not cross (since the tetrahedron does not have opposing vertices).|name=|group=}} and the 32 remaining edges divide the surface into 24 square faces and 8 cubic cells: a [[W:Tesseract|tesseract]]. There are three ways you can do this (choose a set of 8 orthogonal vertices out of 24), so there are three such tesseracts inscribed in the 24-cell.{{Efn|name=three 8-cells}} They overlap with each other, but most of their element sets are disjoint: they share some vertex count, but no edge length, face area, or cell volume.{{Efn|name=vertex-bonded octahedra}} They do share 4-content, their common core.{{Efn||name=common core|group=}}

===== 16-cell =====
Starting with a complete 24-cell, remove the 16 vertices of a tesseract (retaining the 8 vertices you removed above), by cutting through 16 tetrahedral cells bounded by {{sqrt|2}} chords to remove 16 [[W:Tetrahedral pyramid|tetrahedral pyramid]]s whose apexes are the vertices to be removed. This removes 12 great squares (retaining just one orthogonal set) and all the {{sqrt|1}} edges, exposing {{sqrt|2}} chords as the new edges. Now the remaining 6 great squares cross perpendicularly, 3 at each of 8 remaining vertices,{{Efn|The 24-cell's cubical vertex figure{{Efn|name=full size vertex figure}} has been truncated to an octahedral vertex figure. The vertex cube has vanished, and now there are only 6 corners of the vertex figure where before there were 8. The 6 {{sqrt|2}} chords which formerly converged from cube face centers now converge from octahedron vertices; but just as before, they meet at the center where 3 straight lines cross perpendicularly. The octahedron vertices are located 90° away outside the vanished cube, at the new nearest vertices; before truncation those were 24-cell vertices in the second shell of surrounding vertices.|name=|group=}} and their 24 edges divide the surface into 32 triangular faces and 16 tetrahedral cells: a [[16-cell]]. There are three ways you can do this (remove 1 of 3 sets of tesseract vertices), so there are three such 16-cells inscribed in the 24-cell.{{Efn|name=three isoclinic 16-cells}} They overlap with each other, but all of their element sets are disjoint:{{Efn|name=completely disjoint}} they do not share any vertex count, edge length,{{Efn|name=root 2 chords}} or face area, but they do share cell volume. They also share 4-content, their common core.{{Efn||name=common core|group=}}

==== Tetrahedral constructions ====
The 24-cell can be constructed radially from 96 equilateral triangles of edge length {{sqrt|1}} which meet at the center of the polytope, each contributing two radii and an edge. They form 96 {{sqrt|1}} tetrahedra (each contributing one 24-cell face), all sharing the 25th central apex vertex. These form 24 octahedral pyramids (half-16-cells) with their apexes at the center.

The 24-cell can be constructed from 96 equilateral triangles of edge length {{sqrt|2}}, where the three vertices of each triangle are  located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away from each other on the 3-sphere. They form 48 {{sqrt|2}}-edge tetrahedra (the cells of the [[#16-cell|three 16-cells]]), centered at the 24 mid-edge-radii of the 24-cell.{{Efn|Each of the 72 {{sqrt|2}} chords in the 24-cell is a face diagonal in two distinct cubical cells (of different 8-cells) and an edge of four tetrahedral cells (in just one 16-cell).|name=root 2 chords}}

The 24-cell can be constructed directly from its [[#Characteristic orthoscheme|characteristic simplex]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, the [[5-cell#Irregular 5-cells|irregular 5-cell]] which is the [[W:Fundamental region|fundamental region]] of its [[W:Coxeter group|symmetry group]] [[W:F4 polytope|F&lt;sub&gt;4&lt;/sub&gt;]], by reflection of that 4-[[W:Orthoscheme|orthoscheme]] in its own cells (which are 3-orthoschemes).{{Efn|An [[W:Orthoscheme|orthoscheme]] is a [[W:chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:Facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be dissected radially into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring.|name=characteristic orthoscheme}}

==== Cubic constructions ====
The 24-cell is not only the 24-octahedral-cell, it is also the 24-cubical-cell, although the cubes are cells of the three 8-cells, not cells of the 24-cell, in which they are not volumetrically disjoint.

The 24-cell can be constructed from 24 cubes of its own edge length (three 8-cells).{{Efn|name=three 8-cells}} Each of the cubes is shared by 2 8-cells, each of the cubes' square faces is shared by 4 cubes (in 2 8-cells), each of the 96 edges is shared by 8 square faces (in 4 cubes in 2 8-cells), and each of the 96 vertices is shared by 16 edges (in 8 square faces in 4 cubes in 2 8-cells).

== Relationships among interior polytopes ==
The 24-cell, three tesseracts, and three 16-cells are deeply entwined around their common center, and intersect in a common core.{{Efn|The common core of the 24-cell and its inscribed 8-cells and 16-cells is the unit-radius 24-cell's insphere-inscribed dual 24-cell of edge length and radius 1/2.{{Sfn|Coxeter|1995|p=29|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|ps=; &quot;The common content of the 4-cube and the 16-cell is a smaller {3,4,3} whose vertices are the permutations of [(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, 0, 0)]&quot;.}} Rectifying any of the three 16-cells reveals this smaller 24-cell, which has a 4-content of only 1/8 (1/16 that of the unit-radius 24-cell). Its vertices lie at the centers of the 24-cell's octahedral cells, which are also the centers of the tesseracts' square faces, and are also the centers of the 16-cells' edges.{{Sfn|Coxeter|1973|p=147|loc=§8.1 The simple truncations of the general regular polytope|ps=; &quot;At a point of contact, [elements of a regular polytope and elements of its dual in which it is inscribed in some manner] lie in [[W:completely orthogonal|completely orthogonal]] subspaces of the tangent hyperplane to the sphere [of reciprocation], so their only common point is the point of contact itself....{{Efn|name=how planes intersect}} In fact, the [various] radii &lt;sub&gt;0&lt;/sub&gt;𝑹, &lt;sub&gt;1&lt;/sub&gt;𝑹, &lt;sub&gt;2&lt;/sub&gt;𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈&lt;sub&gt;0&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;1&lt;/sub&gt;, 𝐈𝐈&lt;sub&gt;2&lt;/sub&gt;, ... of the original polytope.&quot;}}|name=common core|group=}} The tesseracts and the 16-cells are rotated 60° isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other. This means that the corresponding vertices of two tesseracts or two 16-cells are {{radic|3}} (120°) apart.{{Efn|The 24-cell contains 3 distinct 8-cells (tesseracts), rotated 60° isoclinically with respect to each other. The corresponding vertices of two 8-cells are {{radic|3}} (120°) apart. Each 8-cell contains 8 cubical cells, and each cube contains four {{radic|3}} chords (its long diameters). The 8-cells are not completely disjoint (they share vertices),{{Efn|name=completely disjoint}} but each {{radic|3}} chord occurs as a cube long diameter in just one 8-cell. The {{radic|3}} chords joining the corresponding vertices of two 8-cells belong to the third 8-cell as cube diameters.{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}}|name=three 8-cells}}

The tesseracts are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used twice, are the vertices of three 16-vertex tesseracts.|name=|group=}} such that their vertices and edges are exterior elements of the 24-cell, but their square faces and cubical cells lie inside the 24-cell (they are not elements of the 24-cell). The 16-cells are inscribed in the 24-cell{{Efn|The 24 vertices of the 24-cell, each used once, are the vertices of three 8-vertex 16-cells.{{Efn|name=three basis 16-cells}}|name=|group=}} such that only their vertices are exterior elements of the 24-cell: their edges, triangular faces, and tetrahedral cells lie inside the 24-cell. The interior{{Efn|The edges of the 16-cells are not shown in any of the renderings in this article; if we wanted to show interior edges, they could be drawn as dashed lines. The edges of the inscribed tesseracts are always visible, because they are also edges of the 24-cell.}} 16-cell edges have length {{sqrt|2}}.[[File:Kepler's tetrahedron in cube.png|thumb|Kepler's drawing of tetrahedra in the cube.{{Sfn|Kepler|1619|p=181}}]]

The 16-cells are also inscribed in the tesseracts: their {{sqrt|2}} edges are the face diagonals of the tesseract, and their 8 vertices occupy every other vertex of the tesseract. Each tesseract has two 16-cells inscribed in it (occupying the opposite vertices and face diagonals), so each 16-cell is inscribed in two of the three 8-cells.{{Sfn|van Ittersum|2020|loc=§4.2|pp=73-79}}{{Efn|name=three 16-cells form three tesseracts}} This is reminiscent of the way, in 3 dimensions, two opposing regular tetrahedra can be inscribed in a cube, as discovered by Kepler.{{Sfn|Kepler|1619|p=181}} In fact it is the exact dimensional analogy (the [[W:Demihypercube|demihypercube]]s), and the 48 tetrahedral cells are inscribed in the 24 cubical cells in just that way.{{Sfn|Coxeter|1973|p=269|loc=§14.32|ps=. &quot;For instance, in the case of &lt;math&gt;\gamma_4[2\beta_4]&lt;/math&gt;....&quot;}}{{Efn|name=root 2 chords}}

The 24-cell encloses the three tesseracts within its envelope of octahedral facets, leaving 4-dimensional space in some places between its envelope and each tesseract's envelope of cubes. Each tesseract encloses two of the three 16-cells, leaving 4-dimensional space in some places between its envelope and each 16-cell's envelope of tetrahedra. Thus there are measurable{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions|ps=; An invaluable table providing all 20 metrics of each 4-polytope in edge length units. They must be algebraically converted to compare polytopes of unit radius.}} 4-dimensional interstices{{Efn|The 4-dimensional content of the unit edge length tesseract is 1 (by definition). The content of the unit edge length 24-cell is 2, so half its content is inside each tesseract, and half is between their envelopes. Each 16-cell (edge length {{sqrt|2}}) encloses a content of 2/3, leaving 1/3 of an enclosing tesseract between their envelopes.|name=|group=}} between the 24-cell, 8-cell and 16-cell envelopes. The shapes filling these gaps are [[W:Hyperpyramid|4-pyramids]], alluded to above.{{Efn|Between the 24-cell envelope and the 8-cell envelope, we have the 8 cubic pyramids of Gosset's construction. Between the 8-cell envelope and the 16-cell envelope, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract.}}

== Boundary cells ==
Despite the 4-dimensional interstices between 24-cell, 8-cell and 16-cell envelopes, their 3-dimensional volumes overlap. The different envelopes are separated in some places, and in contact in other places (where no 4-pyramid lies between them). Where they are in contact, they merge and share cell volume: they are the same 3-membrane in those places, not two separate but adjacent 3-dimensional layers.{{Efn|Because there are three overlapping tesseracts inscribed in the 24-cell,{{Efn|name=three 8-cells}} each octahedral cell lies ''on'' a cubic cell of one tesseract (in the cubic pyramid based on the cube, but not in the cube's volume), and ''in'' two cubic cells of each of the other two tesseracts (cubic cells which it spans, sharing their volume).{{Efn|name=octahedral diameters}}|name=octahedra both on and in cubes}} Because there are a total of 7 envelopes, there are places where several envelopes come together and merge volume, and also places where envelopes interpenetrate (cross from inside to outside each other).

Some interior features lie within the 3-space of the (outer) boundary envelope of the 24-cell itself: each octahedral cell is bisected by three perpendicular squares (one from each of the tesseracts), and the diagonals of those squares (which cross each other perpendicularly at the center of the octahedron) are 16-cell edges (one from each 16-cell). Each square bisects an octahedron into two square pyramids, and also bonds two adjacent cubic cells of a tesseract together as their common face.{{Efn|Consider the three perpendicular {{sqrt|2}} long diameters of the octahedral cell.{{Sfn|van Ittersum|2020|p=79}} Each of them is an edge of a different 16-cell. Two of them are the face diagonals of the square face between two cubes; each is a {{sqrt|2}} chord that connects two vertices of those 8-cell cubes across a square face, connects two vertices of two 16-cell tetrahedra (inscribed in the cubes), and connects two opposite vertices of a 24-cell octahedron (diagonally across two of the three orthogonal square central sections).{{Efn|name=root 2 chords}} The third perpendicular long diameter of the octahedron does exactly the same (by symmetry); so it also connects two vertices of a pair of cubes across their common square face: but a different pair of cubes, from one of the other tesseracts in the 24-cell.{{Efn|name=vertex-bonded octahedra}}|name=octahedral diameters}}

As we saw [[#Relationships among interior polytopes|above]], 16-cell {{sqrt|2}} tetrahedral cells are inscribed in tesseract {{sqrt|1}} cubic cells, sharing the same volume. 24-cell {{sqrt|1}} octahedral cells overlap their volume with {{sqrt|1}} cubic cells: they are bisected by a square face into two square pyramids,{{sfn|Coxeter|1973|page=150|postscript=: &quot;Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the &lt;math&gt;\gamma_4&lt;/math&gt;. (Their centres are the mid-points of the 24 edges of the &lt;math&gt;\beta_4&lt;/math&gt;.)&quot;}} the apexes of which also lie at a vertex of a cube.{{Efn|This might appear at first to be angularly impossible, and indeed it would be in a flat space of only three dimensions. If two cubes rest face-to-face in an ordinary 3-dimensional space (e.g. on the surface of a table in an ordinary 3-dimensional room), an octahedron will fit inside them such that four of its six vertices are at the four corners of the square face between the two cubes; but then the other two octahedral vertices will not lie at a cube corner (they will fall within the volume of the two cubes, but not at a cube vertex). In four dimensions, this is no less true! The other two octahedral vertices do ''not'' lie at a corner of the adjacent face-bonded cube in the same tesseract. However, in the 24-cell there is not just one inscribed tesseract (of 8 cubes), there are three overlapping tesseracts (of 8 cubes each). The other two octahedral vertices ''do'' lie at the corner of a cube: but a cube in another (overlapping) tesseract.{{Efn|name=octahedra both on and in cubes}}}} The octahedra share volume not only with the cubes, but with the tetrahedra inscribed in them; thus the 24-cell, tesseracts, and 16-cells all share some boundary volume.{{Efn|name=octahedra both on and in cubes}}

== Radially equilateral honeycomb ==
The dual tessellation of the [[W:24-cell honeycomb|24-cell honeycomb {3,4,3,3}]] is the [[W:16-cell honeycomb|16-cell honeycomb {3,3,4,3}]]. The third regular tessellation of four dimensional space is the [[W:Tesseractic honeycomb|tesseractic honeycomb {4,3,3,4}]], whose vertices can be described by 4-integer Cartesian coordinates.{{Efn|name=quaternions}} The congruent relationships among these three tessellations can be helpful in visualizing the 24-cell, in particular the radial equilateral symmetry which it shares with the tesseract.

A honeycomb of unit edge length 24-cells may be overlaid on a honeycomb of unit edge length tesseracts such that every vertex of a tesseract (every 4-integer coordinate) is also the vertex of a 24-cell (and tesseract edges are also 24-cell edges), and every center of a 24-cell is also the center of a tesseract.{{Sfn|Coxeter|1973|p=163|ps=: Coxeter notes that [[W:Thorold Gosset|Thorold Gosset]] was apparently the first to see that the cells of the 24-cell honeycomb {3,4,3,3} are concentric with alternate cells of the tesseractic honeycomb {4,3,3,4}, and that this observation enabled Gosset's method of construction of the complete set of regular polytopes and honeycombs.}} The 24-cells are twice as large as the tesseracts by 4-dimensional content (hypervolume), so overall there are two tesseracts for every 24-cell, only half of which are inscribed in a 24-cell. If those tesseracts are colored black, and their adjacent tesseracts (with which they share a cubical facet) are colored red, a 4-dimensional checkerboard results.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} Of the 24 center-to-vertex radii{{Efn|It is important to visualize the radii only as invisible interior features of the 24-cell (dashed lines), since they are not edges of the honeycomb. Similarly, the center of the 24-cell is empty (not a vertex of the honeycomb).}} of each 24-cell, 16 are also the radii of a black tesseract inscribed in the 24-cell. The other 8 radii extend outside the black tesseract (through the centers of its cubical facets) to the centers of the 8 adjacent red tesseracts. Thus the 24-cell honeycomb and the tesseractic honeycomb coincide in a special way: 8 of the 24 vertices of each 24-cell do not occur at a vertex of a tesseract (they occur at the center of a tesseract instead). Each black tesseract is cut from a 24-cell by truncating it at these 8 vertices, slicing off 8 cubic pyramids (as in reversing Gosset's construction,{{Sfn|Coxeter|1973|p=150|loc=Gosset}} but instead of being removed the pyramids are simply colored red and left in place). Eight 24-cells meet at the center of each red tesseract: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell. &lt;!-- illustration needed: the red/black checkerboard of the combined 24-cell honeycomb and tesseractic honeycomb; use a vertex-first projection of the 24-cells, and outline the edges of the rhombic dodecahedra as blue lines --&gt;

The red tesseracts are filled cells (they contain a central vertex and radii); the black tesseracts are empty cells. The vertex set of this union of two honeycombs includes the vertices of all the 24-cells and tesseracts, plus the centers of the red tesseracts. Adding the 24-cell centers (which are also the black tesseract centers) to this honeycomb yields a 16-cell honeycomb, the vertex set of which includes all the vertices and centers of all the 24-cells and tesseracts. The formerly empty centers of adjacent 24-cells become the opposite vertices of a unit edge length 16-cell. 24 half-16-cells (octahedral pyramids) meet at each formerly empty center to fill each 24-cell, and their octahedral bases are the 6-vertex octahedral facets of the 24-cell (shared with an adjacent 24-cell).{{Efn|Unlike the 24-cell and the tesseract, the 16-cell is not radially equilateral; therefore 16-cells of two different sizes (unit edge length versus unit radius) occur in the unit edge length honeycomb. The twenty-four 16-cells that meet at the center of each 24-cell have unit edge length, and radius {{sfrac|{{radic|2}}|2}}. The three 16-cells inscribed in each 24-cell have edge length {{radic|2}}, and unit radius.}}

Notice the complete absence of pentagons anywhere in this union of three honeycombs. Like the 24-cell, 4-dimensional Euclidean space itself is entirely filled by a complex of all the polytopes that can be built out of regular triangles and squares (except the 5-cell), but that complex does not require (or permit) any of the pentagonal polytopes.{{Efn|name=pentagonal polytopes}}

== Rotations ==
[[File:24-cell-3CP.gif|thumb|The 24-point 24-cell contains three 8-point 16-cells (red, green, and blue), double-rotated by 60 degrees with respect to each other.{{Sfn|Egan|2019|ps=; Double-rotating 24-cell with orthogonal red, green and blue vertices.}} Each 8-point 16-cell is a coordinate system basis frame of four perpendicular (w,x,y,z) axes.{{Efn|name=three basis 16-cells}} One octahedral cell of the 24 cells is emphasized. Each octahedral cell has two vertices of each color, delimiting an invisible perpendicular axis of the octahedron, which is a {{radic|2}} edge of the red, green, or blue 16-cell.{{Efn|name=octahedral diameters}}]]
The [[#The 24-cell in the proper sequence of 4-polytopes|regular convex 4-polytopes]] are an [[W:Group action|expression]] of their underlying [[W:Symmetry (geometry)|symmetry]] which is known as [[W:SO(4)|SO(4)]],{{Sfn|Goucher|2019|loc=Spin Groups}} the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 24-cell has 1152 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝐹&lt;sub&gt;4&lt;/sub&gt;.}} about a fixed point in 4-dimensional Euclidean space.{{Efn|[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] may occur around a plane, as when adjacent cells are folded around their plane of intersection (by analogy to the way adjacent faces are folded around their line of intersection).{{Efn|Three dimensional [[W:Rotation (mathematics)#In Euclidean geometry|rotations]] occur around an axis line. [[W:Rotations in 4-dimensional Euclidean space|Four dimensional rotations]] may occur around a plane. So in three dimensions we may fold planes around a common line (as when folding a flat net of 6 squares up into a cube), and in four dimensions we may fold cells around a common plane (as when [[W:Tesseract#Geometry|folding a flat net of 8 cubes up into a tesseract]]). Folding around a square face is just folding around ''two'' of its orthogonal edges ''at the same time''; there is not enough space in three dimensions to do this, just as there is not enough space in two dimensions to fold around a line (only enough to fold around a point).|name=simple rotations|group=}} But in four dimensions there is yet another way in which rotations can occur, called a '''[[W:Rotations in 4-dimensional Euclidean space#Geometry of 4D rotations|double rotation]]'''. Double rotations are an emergent phenomenon in the fourth dimension and have no analogy in three dimensions: folding up square faces and folding up cubical cells are both examples of '''simple rotations''', the only kind that occur in fewer than four dimensions. In 3-dimensional rotations, the points in a line remain fixed during the rotation, while every other point moves. In 4-dimensional simple rotations, the points in a plane remain fixed during the rotation, while every other point moves. ''In 4-dimensional double rotations, a point remains fixed during rotation, and every other point moves'' (as in a 2-dimensional rotation!).{{Efn|There are (at least) two kinds of correct [[W:Four-dimensional space#Dimensional analogy|dimensional analogies]]: the usual kind between dimension ''n'' and dimension ''n'' + 1, and the much rarer and less obvious kind between dimension ''n'' and dimension ''n'' + 2. An example of the latter is that rotations in 4-space may take place around a single point, as do rotations in 2-space. Another is the [[W:n-sphere#Other relations|''n''-sphere rule]] that the ''surface area'' of the sphere embedded in ''n''+2 dimensions is exactly 2''π r'' times the ''volume'' enclosed by the sphere embedded in ''n'' dimensions, the most well-known examples being that the circumference of a circle is 2''π r'' times 1, and the surface area of the ordinary sphere is 2''π r'' times 2''r''. Coxeter cites{{Sfn|Coxeter|1973|p=119|loc=§7.1. Dimensional Analogy|ps=: &quot;For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''&lt;sup&gt;2&lt;/sup&gt;, ... it is unlikely that the use of analogy, unaided by computation, would ever lead us to the correct expression [for the hyper-surface of a hyper-sphere], 2''π'' &lt;sup&gt;2&lt;/sup&gt;''r'' &lt;sup&gt;3&lt;/sup&gt;.&quot;}} this as an instance in which dimensional analogy can fail us as a method, but it is really our failure to recognize whether a one- or two-dimensional analogy is the appropriate method.|name=two-dimensional analogy}}|name=double rotations}}

=== The 3 Cartesian bases of the 24-cell ===
There are three distinct orientations of the tesseractic honeycomb which could be made to coincide with the 24-cell [[#Radially equilateral honeycomb|honeycomb]], depending on which of the 24-cell's three disjoint sets of 8 orthogonal vertices (which set of 4 perpendicular axes, or equivalently, which inscribed basis 16-cell){{Efn|name=three basis 16-cells}} was chosen to align it, just as three tesseracts can be inscribed in the 24-cell, rotated with respect to each other.{{Efn|name=three 8-cells}} The distance from one of these orientations to another is an [[#Isoclinic rotations|isoclinic rotation]] through 60 degrees (a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] of 60 degrees in each pair of orthogonal invariant planes, around a single fixed point).{{Efn|name=Clifford displacement}} This rotation can be seen most clearly in the hexagonal central planes, where every hexagon rotates to change which of its three diameters is aligned with a coordinate system axis.{{Efn|name=non-orthogonal hexagons|group=}}

=== Planes of rotation ===
[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes.{{Sfn|Kim|Rote|2016|p=6|loc=§5. Four-Dimensional Rotations}} Thus the general rotation in 4-space is a ''double rotation''.{{Sfn|Perez-Gracia|Thomas|2017|loc=§7. Conclusions|ps=; &quot;Rotations in three dimensions are determined by a rotation axis and the rotation angle about it, where the rotation axis is perpendicular to the plane in which points are being rotated. The situation in four dimensions is more complicated. In this case, rotations are determined by two orthogonal planes
and two angles, one for each plane. Cayley proved that a general 4D rotation can always be decomposed into two 4D rotations, each of them being determined by two equal rotation angles up to a sign change.&quot;}} There are two important special cases, called a ''simple rotation'' and an ''isoclinic rotation''.{{Efn|A [[W:Rotations in 4-dimensional Euclidean space|rotation in 4-space]] is completely characterized by choosing an invariant plane and an angle and direction (left or right) through which it rotates, and another angle and direction through which its one completely orthogonal invariant plane rotates. Two rotational displacements are identical if they have the same pair of invariant planes of rotation, through the same angles in the same directions (and hence also the same chiral pairing of directions). Thus the general rotation in 4-space is a '''double rotation''', characterized by ''two'' angles. A '''simple rotation''' is a special case in which one rotational angle is 0.{{Efn|Any double rotation (including an isoclinic rotation) can be seen as the composition of two simple rotations ''a'' and ''b'': the ''left'' double rotation as ''a'' then ''b'', and the ''right'' double rotation as ''b'' then ''a''. Simple rotations are not commutative; left and right rotations (in general) reach different destinations. The difference between a double rotation and its two composing simple rotations is that the double rotation is 4-dimensionally diagonal: each moving vertex reaches its destination ''directly'' without passing through the intermediate point touched by ''a'' then ''b'', or the other intermediate point touched by ''b'' then ''a'', by rotating on a single helical geodesic (so it is the shortest path).{{Efn|name=helical geodesic}} Conversely, any simple rotation can be seen as the composition of two ''equal-angled'' double rotations (a left isoclinic rotation and a right isoclinic rotation),{{Efn|name=one true circle}} as discovered by [[W:Arthur Cayley|Cayley]]; perhaps surprisingly, this composition ''is'' commutative, and is possible for any double rotation as well.{{Sfn|Perez-Gracia|Thomas|2017}}|name=double rotation}} An '''isoclinic rotation''' is a different special case,{{Efn|name=Clifford displacement}} similar but not identical to two simple rotations through the ''same'' angle.{{Efn|name=plane movement in rotations}}|name=identical rotations}}

==== Simple rotations ====
[[Image:24-cell.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]In 3 dimensions a spinning polyhedron has a single invariant central ''plane of rotation''. The plane is an [[W:Invariant set|invariant set]] because each point in the plane moves in a circle but stays within the plane. Only ''one'' of a polyhedron's central planes can be invariant during a particular rotation; the choice of invariant central plane, and the angular distance and direction it is rotated, completely specifies the rotation. Points outside the invariant plane also move in circles (unless they are on the fixed ''axis of rotation'' perpendicular to the invariant plane), but the circles do not lie within a [[#Geodesics|''central'' plane]].

When a 4-polytope is rotating with only one invariant central plane, the same kind of [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] is happening that occurs in 3 dimensions. One difference is that instead of a fixed axis of rotation, there is an entire fixed central plane in which the points do not move. The fixed plane is the one central plane that is [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} to the invariant plane of rotation. In the 24-cell, there is a simple rotation which will take any vertex ''directly'' to any other vertex, also moving most of the other vertices but leaving at least 2 and at most 6 other vertices fixed (the vertices that the fixed central plane intersects). The vertex moves along a great circle in the invariant plane of rotation between adjacent vertices of a great hexagon, a great square or a great [[W:Digon|digon]], and the completely orthogonal fixed plane is a digon, a square or a hexagon, respectively. {{Efn|In the 24-cell each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane, and each great hexagon plane is completely orthogonal to a plane which intersects only two antipodal vertices: a great [[W:Digon|digon]] plane.|name=pairs of completely orthogonal planes}}

==== Double rotations ====
[[Image:24-cell-orig.gif|thumb|A 3D projection of a 24-cell performing a [[W:SO(4)#Geometry of 4D rotations|double rotation]].{{Sfn|Hise|2007|ps=; Illustration created by Jason Hise with Maya and Macromedia Fireworks.}}]]The points in the completely orthogonal central plane are not ''constrained'' to be fixed. It is also possible for them to be rotating in circles, as a second invariant plane, at a rate independent of the first invariant plane's rotation: a [[W:Rotations in 4-dimensional Euclidean space#Double rotations|double rotation]] in two perpendicular non-intersecting planes{{Efn|name=how planes intersect at a single point}} of rotation at once.{{Efn|name=double rotation}} In a double rotation there is no fixed plane or axis: every point moves except the center point. The angular distance rotated may be different in the two completely orthogonal central planes, but they are always both invariant: their circularly moving points remain within the plane ''as the whole plane tilts sideways'' in the completely orthogonal rotation. A rotation in 4-space always has (at least) ''two'' completely orthogonal invariant planes of rotation, although in a simple rotation the angle of rotation in one of them is 0.

Double rotations come in two [[W:Chiral|chiral]] forms: ''left'' and ''right'' rotations.{{Efn|The adjectives ''left'' and ''right'' are commonly used in two different senses, to distinguish two distinct kinds of pairing. They can refer to alternate directions: the hand on the left side of the body, versus the hand on the right side. Or they can refer to a [[W:Chiral|chiral]] pair of enantiomorphous objects: a left hand is the mirror image of a right hand (like an inside-out glove). In the case of hands the sense intended is rarely ambiguous, because of course the hand on your left side ''is'' the mirror image of the hand on your right side: a hand is either left ''or'' right in both senses. But in the case of double-rotating 4-dimensional objects, only one sense of left versus right properly applies: the enantiomorphous sense, in which the left and right rotation are inside-out mirror images of each other. There ''are'' two directions, which we may call positive and negative, in which moving vertices may be circling on their isoclines, but it would be ambiguous to label those circular directions &quot;right&quot; and &quot;left&quot;, since a rotation's direction and its chirality are independent properties: a right (or left) rotation may be circling in either the positive or negative direction. The left rotation is not rotating &quot;to the left&quot;, the right rotation is not rotating &quot;to the right&quot;, and unlike your left and right hands, double rotations do not lie on the left or right side of the 4-polytope. If double rotations must be analogized to left and right hands, they are better thought of as a pair of clasped hands, centered on the body, because of course they have a common center.|name=clasped hands}} In a double rotation each vertex moves in a spiral along two orthogonal great circles at once.{{Efn|In a double rotation each vertex can be said to move along two completely orthogonal great circles at the same time, but it does not stay within the central plane of either of those original great circles; rather, it moves along a helical geodesic that traverses diagonally between great circles. The two completely orthogonal planes of rotation are said to be ''invariant'' because the points in each stay in their places in the plane ''as the plane moves'', rotating ''and'' tilting sideways by the angle that the ''other'' plane rotates.|name=helical geodesic}} Either the path is right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the &quot;same&quot; directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are &quot;opposite&quot; directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes).{{Sfn|Perez-Gracia|Thomas|2017|loc=§5. A useful mapping|pp=12−13}}

In double rotations of the 24-cell that take vertices to vertices, one invariant plane of rotation contains either a great hexagon, a great square, or only an axis (two vertices, a great digon). The completely orthogonal invariant plane of rotation will necessarily contain a great digon, a great square, or a great hexagon, respectively. The selection of an invariant plane of rotation, a rotational direction and angle through which to rotate it, and a rotational direction and angle through which to rotate its completely orthogonal plane, completely determines the nature of the rotational displacement. In the 24-cell there are several noteworthy kinds of double rotation permitted by these parameters.{{Sfn|Coxeter|1995|loc=(Paper 3) ''Two aspects of the regular 24-cell in four dimensions''|pp=30-32|ps=; §3. The Dodecagonal Aspect;{{Efn|name=Petrie dodecagram and Clifford hexagram}} Coxeter considers the 150°/30° double rotation of period 12 which locates 12 of the 225 distinct 24-cells inscribed in the [[120-cell]], a regular 4-polytope with 120 dodecahedral cells that is the convex hull of the compound of 25 disjoint 24-cells.}}

==== Isoclinic rotations ====
When the angles of rotation in the two invariant planes are exactly the same, a [[W:Rotations in 4-dimensional Euclidean space#Special property of SO(4) among rotation groups in general|remarkably symmetric]] [[W:Geometric transformation|transformation]] occurs:{{Sfn|Perez-Gracia|Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} all the great circle planes Clifford parallel{{Efn|name=Clifford parallels}} to the invariant planes become invariant planes of rotation themselves, through that same angle, and the 4-polytope rotates [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] in many directions at once.{{Sfn|Kim|Rote|2016|loc=§6. Angles between two Planes in 4-Space|pp=7-10}} Each vertex moves an equal distance in four orthogonal directions at the same time.{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. All vertices are displaced to a vertex at least two edge lengths away.{{Efn|name=missing the nearest vertices}} For example, when the unit-radius 24-cell rotates isoclinically 60° in a hexagon invariant plane and 60° in its completely orthogonal invariant plane,{{Efn|name=pairs of completely orthogonal planes}} each vertex is displaced to another vertex {{radic|3}} (120°) away, moving {{radic|3/4}} ≈ 0.866 (half the {{radic|3}} chord length) in four orthogonal directions.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the unit-radius 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of a [[5-cell#Orthoschemes|4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the area of the equilateral triangle face of the unit-edge, unit-radius 24-cell. The area of the radial equilateral triangles in a unit-radius radially equilateral polytope is {{radic|3/4}} ≈ 0.866.|name=root 3/4}}|name=isoclinic 4-dimensional diagonal}} In the 24-cell any isoclinic rotation through 60 degrees in a hexagonal plane takes each vertex to a vertex two edge lengths away, rotates ''all 16'' hexagons by 60 degrees, and takes ''every'' great circle polygon (square,{{Efn|In the [[16-cell#Rotations|16-cell]] the 6 orthogonal great squares form 3 pairs of completely orthogonal great circles; each pair is Clifford parallel. In the 24-cell, the 3 inscribed 16-cells lie rotated 60 degrees isoclinically{{Efn|name=isoclinic 4-dimensional diagonal}} with respect to each other; consequently their corresponding vertices are 120 degrees apart on a hexagonal great circle. Pairing their vertices which are 90 degrees apart reveals corresponding square great circles which are Clifford parallel. Each of the 18 square great circles is Clifford parallel not only to one other square great circle in the same 16-cell (the completely orthogonal one), but also to two square great circles (which are completely orthogonal to each other) in each of the other two 16-cells. (Completely orthogonal great circles are Clifford parallel, but not all Clifford parallels are orthogonal.{{Efn|name=only some Clifford parallels are orthogonal}}) A 60 degree isoclinic rotation of the 24-cell in hexagonal invariant planes takes each square great circle to a Clifford parallel (but non-orthogonal) square great circle in a different 16-cell.|name=Clifford parallel squares in the 16-cell and 24-cell}} hexagon or triangle) to a Clifford parallel great circle polygon of the same kind 120 degrees away. An isoclinic rotation is also called a ''Clifford displacement'', after its [[W:William Kingdon Clifford|discoverer]].{{Efn|In a ''[[W:William Kingdon Clifford|Clifford]] displacement'', also known as an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]], all the Clifford parallel{{Efn|name=Clifford parallels}} invariant planes are displaced in four orthogonal directions at once: they are rotated by the same angle, and at the same time they are tilted ''sideways'' by that same angle in the completely orthogonal rotation.{{Efn|name=one true circle}} A [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|Clifford displacement]] is [[W:8-cell#Radial equilateral symmetry|4-dimensionally diagonal]].{{Efn|name=isoclinic 4-dimensional diagonal}} Every plane that is Clifford parallel to one of the completely orthogonal planes (including in this case an entire Clifford parallel bundle of 4 hexagons, but not all 16 hexagons) is invariant under the isoclinic rotation: all the points in the plane rotate in circles but remain in the plane, even as the whole plane tilts sideways.{{Efn|name=plane movement in rotations}} All 16 hexagons rotate by the same angle (though only 4 of them do so invariantly). All 16 hexagons are rotated by 60 degrees, and also displaced sideways by 60 degrees to a Clifford parallel hexagon. All of the other central polygons (e.g. squares) are also displaced to a Clifford parallel polygon 60 degrees away.|name=Clifford displacement}}

The 24-cell in the ''double'' rotation animation appears to turn itself inside out.{{Efn|That a double rotation can turn a 4-polytope inside out is even more noticeable in the [[W:Rotations in 4-dimensional Euclidean space#Double rotations|tesseract double rotation]].}} It appears to, because it actually does, reversing the [[W:Chirality|chirality]] of the whole 4-polytope just the way your bathroom mirror reverses the chirality of your image by a 180 degree reflection. Each 360 degree isoclinic rotation is as if the 24-cell surface had been stripped off like a glove and turned inside out, making a right-hand glove into a left-hand glove (or vice versa).{{Sfn|Coxeter|1973|p=141|loc=§7.x. Historical remarks|ps=; &quot;[[W:August Ferdinand Möbius|Möbius]] realized, as early as 1827, that a four-dimensional rotation would be required to bring two enantiomorphous solids into coincidence. This idea was neatly deployed by [[W:H. G. Wells|H. G. Wells]] in ''The Plattner Story''.&quot;}}

In a simple rotation of the 24-cell in a hexagonal plane, each vertex in the plane rotates first along an edge to an adjacent vertex 60 degrees away. But in an isoclinic rotation in ''two'' completely orthogonal planes one of which is a great hexagon,{{Efn|name=pairs of completely orthogonal planes}} each vertex rotates first to a vertex ''two'' edge lengths away ({{radic|3}} and 120° distant). The double 60-degree rotation's helical geodesics pass through every other vertex, missing the vertices in between.{{Efn|In an isoclinic rotation vertices move diagonally, like the [[W:bishop (chess)|bishop]]s in [[W:Chess|chess]]. Vertices in an isoclinic rotation ''cannot'' reach their orthogonally nearest neighbor vertices{{Efn|name=8 nearest vertices}} by double-rotating directly toward them (and also orthogonally to that direction), because that double rotation takes them diagonally between their nearest vertices, missing them, to a vertex farther away in a larger-radius surrounding shell of vertices,{{Efn|name=nearest isoclinic vertices are {{radic|3}} away in third surrounding shell}} the way bishops are confined to the white or black squares of the [[W:Chessboard|chessboard]] and cannot reach squares of the opposite color, even those immediately adjacent.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 24 cells (and the 24 vertices) of the 24-cell into two disjoint subsets of 12 cells (and 12 vertices), even and odd (or black and white), which shift places among themselves, in a manner dimensionally analogous to the way the [[W:Bishop (chess)|bishops]]' diagonal moves{{Efn|name=missing the nearest vertices}} restrict them to the black or white squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 24 cells (and 24 vertices) into black and white in the same way.{{Sfn|Coxeter|1973|p=156|loc=|ps=: &quot;...the chess-board has an n-dimensional analogue.&quot;}} The rotations of all fibrations of the same kind of great polygon use the same chessboard, which is a convention of the coordinate system based on even and odd coordinates. ''Left and right are not colors:'' in either a left (or right) rotation half the moving vertices are black, running along black isoclines through black vertices, and the other half are white vertices, also rotating among themselves.{{Efn|Chirality and even/odd parity are distinct flavors. Things which have even/odd coordinate parity are '''''black or white:''''' the squares of the [[W:Chessboard|chessboard]],{{Efn|Since it is difficult to color points and lines white, we sometimes use black and red instead of black and white. In particular, isocline chords are sometimes shown as black or red ''dashed'' lines.{{Efn|name=interior features}}|name=black and red}} '''cells''', '''vertices''' and the '''isoclines''' which connect them by isoclinic rotation.{{Efn|name=isoclinic geodesic}} Everything else is '''''black and white:''''' e.g. adjacent '''face-bonded cell pairs''', or '''edges''' and '''chords''' which are black at one end and white at the other. Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''[[#Isoclinic rotations|isoclinic rotations]]''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''[[#Chiral symmetry operations|sets of Clifford parallel great polygon planes]]''',{{Efn|name=completely orthogonal Clifford parallels are special}} '''[[W:Fiber bundle|fiber bundle]]s''' of Clifford parallel circles (whether or not the circles themselves are chiral), and the chiral cell rings of tetrahedra found in the [[16-cell#Helical construction|16-cell]] and [[600-cell#Boerdijk–Coxeter helix rings|600-cell]]. Things which have '''''neither''''' an even/odd parity nor a chirality include all '''edges''' and '''faces''' (shared by black and white cells), '''[[#Geodesics|great circle polygons]]''' and their '''[[W:Hopf fibration|fibration]]s''', and non-chiral cell rings such as the 24-cell's [[24-cell#Cell rings|cell rings of octahedra]]. Some things are associated with '''''both''''' an even/odd parity and a chirality: '''isoclines''' are black or white because they connect vertices which are all of the same color, and they ''act'' as left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves. Each left (or right) rotation traverses an equal number of black and white isoclines.{{Efn|name=Clifford polygon}}|name=left-right versus black-white}}|name=isoclinic chessboard}}|name=black and white}} Things moving diagonally move farther than 1 unit of distance in each movement step ({{radic|2}} on the chessboard, {{radic|3}} in the 24-cell), but at the cost of ''missing'' half the destinations.{{Efn|name=one true circle}} However, in an isoclinic rotation of a rigid body all the vertices rotate at once, so every destination ''will'' be reached by some vertex. Moreover, there is another isoclinic rotation in hexagon invariant planes which does take each vertex to an adjacent (nearest) vertex. A 24-cell can displace each vertex to a vertex 60° away (a nearest vertex) by rotating isoclinically by 30° in two completely orthogonal invariant planes (one of them a hexagon), ''not'' by double-rotating directly toward the nearest vertex (and also orthogonally to that direction), but instead by double-rotating directly toward a more distant vertex (and also orthogonally to that direction). This helical 30° isoclinic rotation takes the vertex 60° to its nearest-neighbor vertex by a ''different path'' than a simple 60° rotation would. The path along the helical isocline and the path along the simple great circle have the same 60° arc-length, but they consist of disjoint sets of points (except for their endpoints, the two vertices). They are both geodesic (shortest) arcs, but on two alternate kinds of geodesic circle. One is doubly curved (through all four dimensions), and one is simply curved (lying in a two-dimensional plane).|name=missing the nearest vertices}} Each {{radic|3}} chord of the helical geodesic{{Efn|Although adjacent vertices on the isoclinic geodesic are a {{radic|3}} chord apart, a point on a rigid body under rotation does not travel along a chord: it moves along an arc between the two endpoints of the chord (a longer distance). In a ''simple'' rotation between two vertices {{radic|3}} apart, the vertex moves along the arc of a hexagonal great circle to a vertex two great hexagon edges away, and passes through the intervening hexagon vertex midway. But in an ''isoclinic'' rotation between two vertices {{radic|3}} apart the vertex moves along a helical arc called an isocline (not a planar great circle),{{Efn|name=isoclinic geodesic}} which does ''not'' pass through an intervening vertex: it misses the vertex nearest to its midpoint.{{Efn|name=missing the nearest vertices}}|name=isocline misses vertex}} crosses between two Clifford parallel hexagon central planes, and lies in another hexagon central plane that intersects them both.{{Efn|Departing from any vertex V&lt;sub&gt;0&lt;/sub&gt; in the original great hexagon plane of isoclinic rotation P&lt;sub&gt;0&lt;/sub&gt;, the first vertex reached V&lt;sub&gt;1&lt;/sub&gt; is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P&lt;sub&gt;1&lt;/sub&gt;. P&lt;sub&gt;1&lt;/sub&gt; is inclined to P&lt;sub&gt;0&lt;/sub&gt; at a 60° angle.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt; lie in the same hyperplane (the same central cuboctahedron) so their other angle of separation is 0.{{Efn|name=two angles between central planes}}}} The second vertex reached V&lt;sub&gt;2&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;1&lt;/sub&gt; along a second {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;2&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;0&lt;/sub&gt;.{{Efn|P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are 60° apart in ''both'' angles of separation.{{Efn|name=two angles between central planes}} Clifford parallel planes are isoclinic (which means they are separated by two equal angles), and their corresponding vertices are all the same distance apart. Although V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''two'' {{radic|3}} chords apart,{{Efn|V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are two {{radic|3}} chords apart on the geodesic path of this rotational isocline, but that is not the shortest geodesic path between them. In the 24-cell, it is impossible for two vertices to be more distant than ''one'' {{radic|3}} chord, unless they are antipodal vertices {{radic|4}} apart.{{Efn|name=Geodesic distance}} V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt; are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;2&lt;/sub&gt;, the isoclinic rotation has gone the long way around the 24-cell over two {{radic|3}} chords to reach a vertex that was only {{radic|1}} away. More generally, isoclines are geodesics because the distance between their successive vertices is the shortest distance between those two vertices in some rotation connecting them, but on the 3-sphere there may be another rotation which is shorter. A path between two vertices along a geodesic is not always the shortest distance between them (even on ordinary great circle geodesics).}} P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V&lt;sub&gt;1&lt;/sub&gt; lies in both intersecting planes P&lt;sub&gt;1&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt;, as V&lt;sub&gt;0&lt;/sub&gt; lies in both P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;1&lt;/sub&gt;. But P&lt;sub&gt;0&lt;/sub&gt; and P&lt;sub&gt;2&lt;/sub&gt; have ''no'' vertices in common; they do not intersect.) The third vertex reached V&lt;sub&gt;3&lt;/sub&gt; is 120 degrees beyond V&lt;sub&gt;2&lt;/sub&gt; along a third {{radic|3}} chord lying in another hexagonal plane P&lt;sub&gt;3&lt;/sub&gt; that is Clifford parallel to P&lt;sub&gt;1&lt;/sub&gt;. V&lt;sub&gt;0&lt;/sub&gt; and V&lt;sub&gt;3&lt;/sub&gt; are adjacent vertices, {{radic|1}} apart.{{Efn|name=skew hexagram}} The three {{radic|3}} chords lie in different 8-cells.{{Efn|name=three 8-cells}} V&lt;sub&gt;0&lt;/sub&gt; to V&lt;sub&gt;3&lt;/sub&gt; is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram&lt;sub&gt;2&lt;/sub&gt; Clifford polygon.{{Efn|name=Clifford polygon}}|name=360 degree geodesic path visiting 3 hexagonal planes}} The {{radic|3}} chords meet at a 60° angle, but since they lie in different planes they form a [[W:Helix|helix]] not a [[#Great triangles|triangle]]. Three {{radic|3}} chords and 360° of rotation takes the vertex to an adjacent vertex, not back to itself. The helix of {{radic|3}} chords closes into a loop only after six {{radic|3}} chords: a 720° rotation twice around the 24-cell{{Efn|An isoclinic rotation by 60° is two simple rotations by 60° at the same time.{{Efn|The composition of two simple 60° rotations in a pair of completely orthogonal invariant planes is a 60° isoclinic rotation in ''four'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of four simple rotations, and all 24 vertices rotate in invariant hexagon planes, versus just 6 vertices in a simple rotation.}} It moves all the vertices 120° at the same time, in various different directions. Six successive diagonal rotational increments, of 60°x60° each, move each vertex through 720° on a Möbius double loop called an ''isocline'', ''twice'' around the 24-cell and back to its point of origin, in the ''same time'' (six rotational units) that it would take a simple rotation to take the vertex ''once'' around the 24-cell on an ordinary great circle.{{Efn|name=double threaded}} The helical double loop 4𝝅 isocline is just another kind of ''single'' full circle, of the same time interval and period (6 chords) as the simple great circle. The isocline is ''one'' true circle,{{Efn|name=4-dimensional great circles}} as perfectly round and geodesic as the simple great circle, even through its chords are {{radic|3}} longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|All 3-sphere isoclines of the same circumference are directly congruent circles.{{Efn|name=not all isoclines are circles}} An ordinary great circle is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than &lt;math&gt;2\pi r&lt;/math&gt; circumference. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic rotation in which the central planes containing its edges are invariant planes of rotation. The 16-cell and 24-cell edge-rotate on isoclines of 4𝝅 circumference. The 600-cell edge-rotates on isoclines of 5𝝅 circumference.|name=isocline circumference}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly congruent.{{Efn|name=Clifford polygon}} Nevertheless, to avoid confusion we always refer to it as an ''isocline'' and reserve the term ''great circle'' for an ordinary great circle in the plane.{{Efn|name=isocline}}|name=one true circle}} on a [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] [[W:Hexagram|hexagram]] with {{radic|3}} edges.{{Efn|name=skew hexagram}} Even though all 24 vertices and all the hexagons rotate at once, a 360 degree isoclinic rotation moves each vertex only halfway around its circuit. After 360 degrees each helix has departed from 3 vertices and reached a fourth vertex adjacent to the original vertex, but has ''not'' arrived back exactly at the vertex it departed from. Each central plane (every hexagon or square in the 24-cell) has rotated 360 degrees ''and'' been tilted sideways all the way around 360 degrees back to its original position (like a coin flipping twice), but the 24-cell's [[W:Orientation entanglement|orientation]] in the 4-space in which it is embedded is now different.{{Sfn|Mebius|2015|loc=Motivation|pp=2-3|ps=; &quot;This research originated from ... the desire to construct a computer implementation of a specific motion of the human arm, known among folk dance experts as the ''Philippine wine dance'' or ''Binasuan'' and performed by physicist [[W:Richard P. Feynman|Richard P. Feynman]] during his [[W:Dirac|Dirac]] memorial lecture 1986{{Sfn|Feynman|Weinberg|1987|loc=The reason for antiparticles}} to show that a single rotation (2𝝅) is not equivalent in all respects to no rotation at all, whereas a double rotation (4𝝅) is.&quot;}} Because the 24-cell is now inside-out, if the isoclinic rotation is continued in the ''same'' direction through another 360 degrees, the 24 moving vertices will pass through the other half of the vertices that were missed on the first revolution (the 12 antipodal vertices of the 12 that were hit the first time around), and each isoclinic geodesic ''will'' arrive back at the vertex it departed from, forming a closed six-chord helical loop. It takes a 720 degree isoclinic rotation for each [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic]] to complete a circuit through every ''second'' vertex of its six vertices by [[W:Winding number|winding]] around the 24-cell twice, returning the 24-cell to its original chiral orientation.{{Efn|In a 720° isoclinic rotation of a ''rigid'' 24-cell the 24 vertices rotate along four separate Clifford parallel hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic loops (six vertices circling in each loop) and return to their original positions.{{Efn|name=Villarceau circles}}}}

The hexagonal winding path that each vertex takes as it loops twice around the 24-cell forms a double helix bent into a [[W:Möbius strip|Möbius ring]], so that the two strands of the double helix form a continuous single strand in a closed loop.{{Efn|Because the 24-cell's helical hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself in each revolution, reversing its chirality{{Efn|name=Clifford polygon}} but without ever changing its even/odd parity of rotation (black or white).{{Efn|name=black and white}} The 6-vertex isoclinic path forms a Möbius double loop, like a 3-dimensional double helix with the ends of its two parallel 3-vertex helices cross-connected to each other. This 60° isocline{{Efn|A strip of paper can form a [[W:Möbius strip#Polyhedral surfaces and flat foldings|flattened Möbius strip]] in the plane by folding it at &lt;math&gt;60^\circ&lt;/math&gt; angles so that its center line lies along an equilateral triangle, and attaching the ends. The shortest strip for which this is possible consists of three equilateral paper triangles, folded at the edges where two triangles meet. Since the loop traverses both sides of each paper triangle, it is a hexagonal loop over six equilateral triangles. Its [[W:Aspect ratio|aspect ratio]]{{snd}}the ratio of the strip's length{{efn|The length of a strip can be measured at its centerline, or by cutting the resulting Möbius strip perpendicularly to its boundary so that it forms a rectangle.}} to its width{{snd}}is {{nowrap|&lt;math&gt;\sqrt 3\approx 1.73&lt;/math&gt;.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram&lt;sub&gt;2&lt;/sub&gt;.{{Efn|name=skew hexagram}} Successive {{radic|3}} edges belong to different [[#8-cell|8-cells]], as the 720° isoclinic rotation takes each hexagon through all six hexagons in the [[#6-cell rings|6-cell ring]], and each 8-cell through all three 8-cells twice.{{Efn|name=three 8-cells}}|name=double threaded}} In the first revolution the vertex traverses one 3-chord strand of the double helix; in the second revolution it traverses the second 3-chord strand, moving in the same rotational direction with the same handedness (bending either left or right) throughout. Although this isoclinic Möbius [[#6-cell rings|ring]] is a circular spiral through all 4 dimensions, not a 2-dimensional circle, like a great circle it is a geodesic because it is the shortest path from vertex to vertex.{{Efn|A point under isoclinic rotation traverses the diagonal{{Efn|name=isoclinic 4-dimensional diagonal}} straight line of a single '''isoclinic geodesic''', reaching its destination directly, instead of the bent line of two successive '''simple geodesics'''.{{Efn||name=double rotation}} A '''[[W:Geodesic|geodesic]]''' is the ''shortest path'' through a space (intuitively, a string pulled taught between two points). Simple geodesics are great circles lying in a central plane (the only kind of geodesics that occur in 3-space on the 2-sphere). Isoclinic geodesics are different: they do ''not'' lie in a single plane; they are 4-dimensional [[W:Helix|spirals]] rather than simple 2-dimensional circles.{{Efn|name=helical geodesic}} But they are not like 3-dimensional [[W:Screw threads|screw threads]] either, because they form a closed loop like any circle.{{Efn|name=double threaded}} Isoclinic geodesics are ''4-dimensional great circles'', and they are just as circular as 2-dimensional circles: in fact, twice as circular, because they curve in ''two'' orthogonal great circles at once.{{Efn|Isoclinic geodesics or ''isoclines'' are 4-dimensional great circles in the sense that they are 1-dimensional geodesic ''lines'' that curve in 4-space in two orthogonal great circles at once.{{Efn|name=not all isoclines are circles}} They should not be confused with ''great 2-spheres'',{{Sfn|Stillwell|2001|p=24}} which are the 4-dimensional analogues of great circles (great 1-spheres).{{Efn|name=great 2-spheres}} Discrete isoclines are polygons;{{Efn|name=Clifford polygon}} discrete great 2-spheres are polyhedra.|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles.{{Efn|name=hexagonal fibrations}}{{Efn|name=square fibrations}} These '''isoclines''' are geodesic 1-dimensional lines embedded in a 4-dimensional space. On the 3-sphere{{Efn|All isoclines are [[W:Geodesics|geodesics]], and isoclines on the [[W:3-sphere|3-sphere]] are circles (curving equally in each dimension), but not all isoclines on 3-manifolds in 4-space are circles.|name=not all isoclines are circles}} they always occur in pairs{{Efn|Isoclines on the 3-sphere occur in non-intersecting pairs of even/odd coordinate parity.{{Efn|name=black and white}} A single black or white isocline forms a [[W:Möbius loop|Möbius loop]] called the {1,1} torus knot or Villarceau circle{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]]. Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it. Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus. By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}} in which each of two &quot;circles&quot; linked in a Möbius &quot;figure eight&quot; loop traverses through all four dimensions.{{Efn|name=Clifford polygon}} The double loop is a true circle in four dimensions.{{Efn|name=one true circle}} Even and odd isoclines are also linked, not in a Möbius loop but as a [[W:Hopf link|Hopf link]] of two non-intersecting circles,{{Efn|name=Clifford parallels}} as are all the Clifford parallel isoclines of a [[W:Hopf fibration|Hopf fiber bundle]].|name=Villarceau circles}} as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]], the geodesic paths traversed by vertices in an [[W:Rotations in 4-dimensional Euclidean space#Double rotations|isoclinic rotation]]. They are [[W:Helix|helices]] bent into a [[W:Möbius strip|Möbius loop]] in the fourth dimension, taking a diagonal [[W:Winding number|winding route]] around the 3-sphere through the non-adjacent vertices{{Efn|name=missing the nearest vertices}} of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}

=== Clifford parallel polytopes ===
Two planes are also called ''isoclinic'' if an isoclinic rotation will bring them together.{{Efn|name=two angles between central planes}} The isoclinic planes are precisely those central planes with Clifford parallel geodesic great circles.{{Sfn|Kim|Rote|2016|loc=Relations to Clifford parallelism|pp=8-9}} Clifford parallel great circles do not intersect,{{Efn|name=Clifford parallels}} so isoclinic great circle polygons have disjoint vertices. In the 24-cell every hexagonal central plane is isoclinic to three others, and every square central plane is isoclinic to five others. We can pick out 4 mutually isoclinic (Clifford parallel) great hexagons (four different ways) covering all 24 vertices of the 24-cell just once (a hexagonal fibration).{{Efn|The 24-cell has four sets of 4 non-intersecting [[W:Clifford parallel|Clifford parallel]]{{Efn|name=Clifford parallels}} great circles each passing through 6 vertices (a great hexagon), with only one great hexagon in each set passing through each vertex, and the 4 hexagons in each set reaching all 24 vertices.{{Efn|name=four hexagonal fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of interlocking great circles. The 24-cell can also be divided (eight different ways) into 4 disjoint subsets of 6 vertices (hexagrams) that do ''not'' lie in a hexagonal central plane, each skew [[#Helical hexagrams and their  isoclines|hexagram forming an isoclinic geodesic or ''isocline'']] that is the rotational circle traversed by those 6 vertices in one particular left or right [[#Isoclinic rotations|isoclinic rotation]]. Each of these sets of four Clifford parallel isoclines belongs to one of the four discrete Hopf fibrations of hexagonal great circles.{{Efn|Each set of [[W:Clifford parallel|Clifford parallel]] [[#Geodesics|great circle]] polygons is a different bundle of fibers than the corresponding set of Clifford parallel isocline{{Efn|name=isoclinic geodesic}} polygrams, but the two [[W:Fiber bundles|fiber bundles]] together constitute the ''same'' discrete [[W:Hopf fibration|Hopf fibration]], because they enumerate the 24 vertices together by their intersection in the same distinct (left or right) isoclinic rotation. They are the [[W:Warp and woof|warp and woof]] of the same woven fabric that is the fibration.|name=great circles and isoclines are same fibration|name=warp and woof}}|name=hexagonal fibrations}} We can pick out 6 mutually isoclinic (Clifford parallel) great squares{{Efn|Each great square plane is isoclinic (Clifford parallel) to five other square planes but [[W:Completely orthogonal|completely orthogonal]] to only one of them.{{Efn|name=Clifford parallel squares in the 16-cell and 24-cell}} Every pair of completely orthogonal planes has Clifford parallel great circles, but not all Clifford parallel great circles are orthogonal (e.g., none of the hexagonal geodesics in the 24-cell are mutually orthogonal). There is also another way in which completely orthogonal planes are in a distinguished category of Clifford parallel planes: they are not [[W:Chiral|chiral]], or strictly speaking they possess both chiralities. A pair of isoclinic (Clifford parallel) planes is either a ''left pair'' or a ''right pair'', unless they are separated by two angles of 90° (completely orthogonal planes) or 0° (coincident planes).{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} Most isoclinic planes are brought together only by a left isoclinic rotation or a right isoclinic rotation, respectively. Completely orthogonal planes are special: the pair of planes is both a left and a right pair, so either a left or a right isoclinic rotation will bring them together. This occurs because isoclinic square planes are 180° apart at all vertex pairs: not just Clifford parallel but completely orthogonal. The isoclines (chiral vertex paths){{Efn|name=isoclinic geodesic}} of 90° isoclinic rotations are special for the same reason. Left and right isoclines loop through the same set of antipodal vertices (hitting both ends of each [[16-cell#Helical construction|16-cell axis]]), instead of looping through disjoint left and right subsets of black or white antipodal vertices (hitting just one end of each axis), as the left and right isoclines of all other fibrations do.|name=completely orthogonal Clifford parallels are special}} (three different ways) covering all 24 vertices of the 24-cell just once (a square fibration).{{Efn|The 24-cell has three sets of 6 non-intersecting Clifford parallel great circles each passing through 4 vertices (a great square), with only one great square in each set passing through each vertex, and the 6 squares in each set reaching all 24 vertices.{{Efn|name=three square fibrations}} Each set constitutes a discrete [[W:Hopf fibration|Hopf fibration]] of 6 interlocking great squares, which is simply the compound of the three inscribed 16-cell's discrete Hopf fibrations of 2 interlocking great squares. The 24-cell can also be divided (six different ways) into 3 disjoint subsets of 8 vertices (octagrams) that do ''not'' lie in a square central plane, but comprise a 16-cell and lie on a skew [[#Helical octagrams and thei isoclines|octagram&lt;sub&gt;3&lt;/sub&gt; forming an isoclinic geodesic or ''isocline'']] that is the rotational cirle traversed by those 8 vertices in one particular left or right [[16-cell#Rotations|isoclinic rotation]] as they rotate positions within the 16-cell.{{Efn|name=warp and woof}}|name=square fibrations}} Every isoclinic rotation taking vertices to vertices corresponds to a discrete fibration.{{Efn|name=fibrations are distinguished only by rotations}}

Two dimensional great circle polygons are not the only polytopes in the 24-cell which are parallel in the Clifford sense.{{Sfn|Tyrrell|Semple|1971|pp=1-9|loc=§1. Introduction}} Congruent polytopes of 2, 3 or 4 dimensions can be said to be Clifford parallel in 4 dimensions if their corresponding vertices are all the same distance apart. The three 16-cells inscribed in the 24-cell are Clifford parallels. Clifford parallel polytopes are ''completely disjoint'' polytopes.{{Efn|Polytopes are '''completely disjoint''' if all their ''element sets'' are disjoint: they do not share any vertices, edges, faces or cells. They may still overlap in space, sharing 4-content, volume, area, or lineage.|name=completely disjoint}} A 60 degree isoclinic rotation in hexagonal planes takes each 16-cell to a disjoint 16-cell. Like all [[#Double rotations|double rotations]], isoclinic rotations come in two [[W:Chiral|chiral]] forms: there is a disjoint 16-cell to the ''left'' of each 16-cell, and another to its ''right''.{{Efn|Visualize the three [[16-cell]]s inscribed in the 24-cell (left, right, and middle), and the rotation which takes them to each other. [[#Reciprocal constructions from 8-cell and 16-cell|The vertices of the middle 16-cell lie on the (w, x, y, z) coordinate axes]];{{Efn|name=Six orthogonal planes of the Cartesian basis}} the other two are rotated 60° [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinically]] to its left and its right. The 24-vertex 24-cell is a compound of three 16-cells, whose three sets of 8 vertices are distributed around the 24-cell symmetrically; each vertex is surrounded by 8 others (in the 3-dimensional space of the 4-dimensional 24-cell's ''surface''), the way the vertices of a cube surround its center.{{Efn|name=24-cell vertex figure}} The 8 surrounding vertices (the cube corners) lie in other 16-cells: 4 in the other 16-cell to the left, and 4 in the other 16-cell to the right. They are the vertices of two tetrahedra inscribed in the cube, one belonging (as a cell) to each 16-cell. If the 16-cell edges are {{radic|2}}, each vertex of the compound of three 16-cells is {{radic|1}} away from its 8 surrounding vertices in other 16-cells. Now visualize those {{radic|1}} distances as the edges of the 24-cell (while continuing to visualize the disjoint 16-cells). The {{radic|1}} edges form great hexagons of 6 vertices which run around the 24-cell in a central plane. ''Four'' hexagons cross at each vertex (and its antipodal vertex), inclined at 60° to each other.{{Efn|name=cuboctahedral hexagons}} The [[#Great hexagons|hexagons]] are not perpendicular to each other, or to the 16-cells' perpendicular [[#Great squares|square central planes]].{{Efn|name=non-orthogonal hexagons}} The left and right 16-cells form a tesseract.{{Efn|Each pair of the three 16-cells inscribed in the 24-cell forms a 4-dimensional [[W:Tesseract|hypercube (a tesseract or 8-cell)]], in [[#Relationships among interior polytopes|dimensional analogy]] to the way two tetrahedra form a cube: the two 8-vertex 16-cells are inscribed in the 16-vertex tesseract, occupying its alternate vertices. The third 16-cell does not lie within the tesseract; its 8 vertices protrude from the sides of the tesseract, forming a cubic pyramid on each of the tesseract's cubic cells (as in [[#Reciprocal constructions from 8-cell and 16-cell|Gosset's construction of the 24-cell]]). The three pairs of 16-cells form three tesseracts.{{Efn|name=three 8-cells}} The tesseracts share vertices, but the 16-cells are completely disjoint.{{Efn|name=completely disjoint}}|name=three 16-cells form three tesseracts}} Two 16-cells have vertex-pairs which are one {{radic|1}} edge (one hexagon edge) apart. But a [[#Simple rotations|''simple'' rotation]] of 60° will not take one whole 16-cell to another 16-cell, because their vertices are 60° apart in different directions, and a simple rotation has only one hexagonal plane of rotation. One 16-cell ''can'' be taken to another 16-cell by a 60° [[#Isoclinic rotations|''isoclinic'' rotation]], because an isoclinic rotation is [[W:3-sphere|3-sphere]] symmetric: four [[#Clifford parallel polytopes|Clifford parallel hexagonal planes]] rotate together, but in four different rotational directions,{{Efn|name=Clifford displacement}} taking each 16-cell to another 16-cell. But since an isoclinic 60°  rotation is a ''diagonal'' rotation by 60° in ''two'' orthogonal great circles at once,{{Efn|name=isoclinic geodesic}} the corresponding vertices of the 16-cell and the 16-cell it is taken to are 120° apart: ''two'' {{radic|1}} hexagon edges (or one {{radic|3}} hexagon chord) apart, not one {{radic|1}} edge (60°) apart.{{Efn|name=isoclinic 4-dimensional diagonal}} By the [[W:Chiral|chiral]] diagonal nature of isoclinic rotations, the 16-cell ''cannot'' reach the adjacent 16-cell (whose vertices are one {{radic|1}} edge away) by rotating toward it;{{Efn|name=missing the nearest vertices}} it can only reach the 16-cell ''beyond'' it (120° away). But of course, the 16-cell beyond the 16-cell to its right is the 16-cell to its left. So a 60° isoclinic  rotation ''will'' take every 16-cell to another 16-cell: a 60° ''right'' isoclinic rotation will take the middle 16-cell to the 16-cell we may have originally visualized as the ''left'' 16-cell, and a 60° ''left'' isoclinic rotation will take the middle 16-cell to the 16-cell we visualized as the ''right'' 16-cell. (If so, that was our error in visualization; the 16-cell to the &quot;left&quot; is in fact the one reached by the left isoclinic rotation, as that is the only [[#Double rotations|sense in which the two 16-cells are left or right]] of each other.){{Efn|name=clasped hands}}|name=three isoclinic 16-cells}}

All Clifford parallel 4-polytopes are related by an isoclinic rotation,{{Efn|name=Clifford displacement}} but not all isoclinic polytopes are Clifford parallels (completely disjoint).{{Efn|All isoclinic ''planes'' are Clifford parallels (completely disjoint).{{Efn|name=completely disjoint}} Three and four dimensional cocentric objects may intersect (sharing elements) but still be related by an isoclinic rotation. Polyhedra and 4-polytopes may be isoclinic and ''not'' disjoint, if all of their corresponding planes are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same plane).}} The three 8-cells in the 24-cell are isoclinic but not Clifford parallel. Like the 16-cells, they are rotated 60 degrees isoclinically with respect to each other, but their vertices are not all disjoint (and therefore not all equidistant). Each vertex occurs in two of the three 8-cells (as each 16-cell occurs in two of the three 8-cells).{{Efn|name=three 8-cells}}

Isoclinic rotations relate the convex regular 4-polytopes to each other. An isoclinic rotation of a single 16-cell will generate{{Efn|By ''generate'' we mean simply that some vertex of the first polytope will visit each vertex of the generated polytope in the course of the rotation.}} a 24-cell. A simple rotation of a single 16-cell will not, because its vertices will not reach either of the other two 16-cells' vertices in the course of the rotation. An isoclinic rotation of the 24-cell will generate the 600-cell, and an isoclinic rotation of the 600-cell will generate the 120-cell. (Or they can all be generated directly by an isoclinic rotation of the 16-cell, generating isoclinic copies of itself.) The different convex regular 4-polytopes nest inside each other, and multiple instances of the same 4-polytope hide next to each other in the Clifford parallel spaces that comprise the 3-sphere.{{Sfn|Tyrrell|Semple|1971|loc=Clifford Parallel Spaces and Clifford Reguli|pp=20-33}} For an object of more than one dimension, the only way to reach these parallel subspaces directly is by isoclinic rotation. Like a key operating a four-dimensional lock, an object must twist in two completely perpendicular tumbler cylinders at once in order to move the short distance between Clifford parallel subspaces.

=== Rings ===
In the 24-cell there are sets of rings of six different kinds, described separately in detail in other sections of [[24-cell|this article]]. This section describes how the different kinds of rings are [[#Relationships among interior polytopes|intertwined]].

The 24-cell contains four kinds of [[#Geodesics|geodesic fibers]] (polygonal rings running through vertices): [[#Great squares|great circle squares]] and their [[16-cell#Helical construction|isoclinic helix octagrams]],{{Efn|name=square fibrations}} and [[#Great hexagons|great circle hexagons]] and their [[#Isoclinic rotations|isoclinic helix hexagrams]].{{Efn|name=hexagonal fibrations}} It also contains two kinds of [[24-cell#Cell rings|cell rings]] (chains of octahedra bent into a ring in the fourth dimension): four octahedra connected vertex-to-vertex and bent into a square, and six octahedra connected face-to-face and bent into a hexagon.{{Sfn|Coxeter|1970|loc=§8. The simplex, cube, cross-polytope and 24-cell|p=18|ps=; Coxeter studied cell rings in the general case of their geometry and [[W:Group theory|group theory]], identifying each cell ring as a [[W:Polytope|polytope]] in its own right which fills a three-dimensional manifold (such as the [[W:3-sphere|3-sphere]]) with its corresponding [[W:Honeycomb (geometry)|honeycomb]]. He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s{{Efn|name=Petrie dodecagram and Clifford hexagram}} and some (but not all) cell rings and their honeycombs are ''twisted'', occurring in left- and right-handed [[chiral]] forms. Specifically, he found that since the 24-cell's octahedral cells have opposing faces, the cell rings in the 24-cell are of the non-chiral (directly congruent) kind.{{Efn|name=6-cell ring is not chiral}} Each of the 24-cell's cell rings has its corresponding honeycomb in Euclidean (rather than hyperbolic) space, so the 24-cell tiles 4-dimensional Euclidean space by translation to form the [[W:24-cell honeycomb|24-cell honeycomb]].}}{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}}

==== 4-cell rings ====
Four unit-edge-length octahedra can be connected vertex-to-vertex along a common axis of length 4{{radic|2}}. The axis can then be bent into a square of edge length {{radic|2}}. Although it is possible to do this in a space of only three dimensions, that is not how it occurs in the 24-cell. Although the {{radic|2}} axes of the four octahedra occupy the same plane, forming one of the 18 {{radic|2}} great squares of the 24-cell, each octahedron occupies a different 3-dimensional hyperplane,{{Efn|Just as each face of a [[W:Polyhedron|polyhedron]] occupies a different (2-dimensional) face plane, each cell of a [[W:Polychoron|polychoron]] occupies a different (3-dimensional) cell [[W:Hyperplane|hyperplane]].{{Efn|name=hyperplanes}}}} and all four dimensions are utilized. The 24-cell can be partitioned into 6 such 4-cell rings (three different ways), mutually interlinked like adjacent links in a chain (but these [[W:Link (knot theory)|links]] all have a common center). An [[#Isoclinic rotations|isoclinic rotation]] in the great square plane by a multiple of 90° takes each octahedron in the ring to an octahedron in the ring.

==== 6-cell rings ====
[[File:Six face-bonded octahedra.jpg|thumb|400px|A 4-dimensional ring of 6 face-bonded octahedra, bounded by two intersecting sets of three Clifford parallel great hexagons of different colors, cut and laid out flat in 3 dimensional space.{{Efn|name=6-cell ring}}]]Six regular octahedra can be connected face-to-face along a common axis that passes through their centers of volume, forming a stack or column with only triangular faces. In a space of four dimensions, the axis can then be bent 60° in the fourth dimension at each of the six octahedron centers, in a plane orthogonal to all three orthogonal central planes of each octahedron, such that the top and bottom triangular faces of the column become coincident. The column becomes a ring around a hexagonal axis. The 24-cell can be partitioned into 4 such rings (four different ways), mutually interlinked. Because the hexagonal axis joins cell centers (not vertices), it is not a great hexagon of the 24-cell.{{Efn|The axial hexagon of the 6-octahedron ring does not intersect any vertices or edges of the 24-cell, but it does hit faces. In a unit-edge-length 24-cell, it has edges of length 1/2.{{Efn|When unit-edge octahedra are placed face-to-face the distance between their centers of volume is {{radic|2/3}} ≈ 0.816.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(i): Octahedron}} When 24 face-bonded octahedra are bent into a 24-cell lying on the 3-sphere, the centers of the octahedra are closer together in 4-space. Within the curved 3-dimensional surface space filled by the 24 cells, the cell centers are still {{radic|2/3}} apart along the curved geodesics that join them. But on the straight chords that join them, which dip inside the 3-sphere, they are only 1/2 edge length apart.}} Because it joins six cell centers, the axial hexagon is a great hexagon of the smaller dual 24-cell that is formed by joining the 24 cell centers.{{Efn|name=common core}}}} However, six great hexagons can be found in the ring of six octahedra, running along the edges of the octahedra. In the column of six octahedra (before it is bent into a ring) there are six spiral paths along edges running up the column: three parallel helices spiraling clockwise, and three parallel helices spiraling counterclockwise. Each clockwise helix intersects each counterclockwise helix at two vertices three edge lengths apart. Bending the column into a ring changes these helices into great circle hexagons.{{Efn|There is a choice of planes in which to fold the column into a ring, but they are equivalent in that they produce congruent rings. Whichever folding planes are chosen, each of the six helices joins its own two ends and forms a simple great circle hexagon. These hexagons are ''not'' helices: they lie on ordinary flat great circles. Three of them are Clifford parallel{{Efn|name=Clifford parallels}} and belong to one [[#Great hexagons|hexagonal]] fibration. They intersect the other three, which belong to another hexagonal fibration. The three parallel great circles of each fibration spiral around each other in the sense that they form a [[W:Link (knot theory)|link]] of three ordinary circles, but they are not twisted: the 6-cell ring has no [[W:Torsion of a curve|torsion]], either clockwise or counterclockwise.{{Efn|name=6-cell ring is not chiral}}|name=6-cell ring}} The ring has two sets of three great hexagons, each on three Clifford parallel great circles.{{Efn|The three great hexagons are Clifford parallel, which is different than ordinary parallelism.{{Efn|name=Clifford parallels}} Clifford parallel great hexagons pass through each other like adjacent links of a chain, forming a [[W:Hopf link|Hopf link]]. Unlike links in a 3-dimensional chain, they share the same center point. In the 24-cell, Clifford parallel great hexagons occur in sets of four, not three. The fourth parallel hexagon lies completely outside the 6-cell ring; its 6 vertices are completely disjoint from the ring's 18 vertices.}} The great hexagons in each parallel set of three do not intersect, but each intersects the other three great hexagons (to which it is not Clifford parallel) at two antipodal vertices.

A [[#Simple rotations|simple rotation]] in any of the great hexagon planes by a multiple of 60° rotates only that hexagon invariantly, taking each vertex in that hexagon to a vertex in the same hexagon. An [[#Isoclinic rotations|isoclinic rotation]] by 60° in any of the six great hexagon planes rotates all three Clifford parallel great hexagons invariantly, and takes each octahedron in the ring to a ''non-adjacent'' octahedron in the ring.{{Efn|An isoclinic rotation by a multiple of 60° takes even-numbered octahedra in the ring to even-numbered octahedra, and odd-numbered octahedra to odd-numbered octahedra.{{Efn|In the column of 6 octahedral cells, we number the cells 0-5 going up the column. We also label each vertex with an integer 0-5 based on how many edge lengths it is up the column.}} It is impossible for an even-numbered octahedron to reach an odd-numbered octahedron, or vice versa, by a left or a right isoclinic rotation alone.{{Efn|name=black and white}}|name=black and white octahedra}}

Each isoclinically displaced octahedron is also rotated itself. After a 360° isoclinic rotation each octahedron is back in the same position, but in a different orientation. In a 720° isoclinic rotation, its vertices are returned to their original [[W:Orientation entanglement|orientation]].

Four Clifford parallel great hexagons comprise a discrete fiber bundle covering all 24 vertices in a [[W:Hopf fibration|Hopf fibration]]. The 24-cell has four such [[#Great hexagons|discrete hexagonal fibrations]] &lt;math&gt;F_a, F_b, F_c, F_d&lt;/math&gt;. Each great hexagon belongs to just one fibration, and the four fibrations are defined by disjoint sets of four great hexagons each.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16|ps=; Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle.}} Each fibration is the domain (container) of a unique left-right pair of isoclinic rotations (left and right Hopf fiber bundles).{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings (a fibration) is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. Each fibration corresponds to a left-right pair of isoclinic rotations in a particular set of Clifford parallel invariant central planes of rotation. In the 24-cell, distinguishing a hexagonal fibration{{Efn|name=hexagonal fibrations}} means choosing a cell-disjoint set of four 6-cell rings that is the unique container of a left-right pair of isoclinic rotations in four Clifford parallel hexagonal invariant planes. The left and right rotations take place in chiral subspaces of that container,{{Sfn|Kim|Rote|2016|p=12|loc=§8 The Construction of Hopf Fibrations; 3}} but the fibration and the octahedral cell rings themselves are not chiral objects.{{Efn|name=6-cell ring is not chiral}}|name=fibrations are distinguished only by rotations}}

Four cell-disjoint 6-cell rings also comprise each discrete fibration defined by four Clifford parallel great hexagons. Each 6-cell ring contains only 18 of the 24 vertices, and only 6 of the 16 great hexagons, which we see illustrated above running along the cell ring's edges: 3 spiraling clockwise and 3 counterclockwise. Those 6 hexagons running along the cell ring's edges are not among the set of four parallel hexagons which define the fibration. For example, one of the four 6-cell rings in fibration &lt;math&gt;F_a&lt;/math&gt; contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration &lt;math&gt;F_b&lt;/math&gt;, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration &lt;math&gt;F_c&lt;/math&gt;, but that cell ring contains no great hexagons from fibration &lt;math&gt;F_a&lt;/math&gt; or fibration &lt;math&gt;F_d&lt;/math&gt;.

The 24-cell contains 16 great hexagons, divided into four disjoint sets of four hexagons, each disjoint set uniquely defining a fibration. Each fibration is also a distinct set of four cell-disjoint 6-cell rings. The 24-cell has exactly 16 distinct 6-cell rings. Each 6-cell ring belongs to just one of the four fibrations.{{Efn|The dual polytope of the 24-cell is another 24-cell. It can be constructed by placing vertices at the 24 cell centers. Each 6-cell ring corresponds to a great hexagon in the dual 24-cell, so there are 16 distinct 6-cell rings, as there are 16 distinct great hexagons, each belonging to just one fibration.}}

==== Helical hexagrams and their isoclines ====
Another kind of geodesic fiber, the [[#Isoclinic rotations|helical hexagram isoclines]], can be found within a 6-cell ring of octahedra. Each of these geodesics runs through every ''second'' vertex of a skew [[W:Hexagram|hexagram]]&lt;sub&gt;2&lt;/sub&gt;, which in the unit-radius, unit-edge-length 24-cell has six {{radic|3}} edges. The hexagram does not lie in a single central plane, but is composed of six linked {{radic|3}} chords from the six different hexagon great circles in the 6-cell ring. The isocline geodesic fiber is the path of an isoclinic rotation,{{Efn|name=isoclinic geodesic}} a helical rather than simply circular path around the 24-cell which links vertices two edge lengths apart and consequently must wrap twice around the 24-cell before completing its six-vertex loop.{{Efn|The chord-path of an isocline (the geodesic along which a vertex moves under isoclinic rotation) may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}} The isocline is a helical Möbius double loop which reverses its chirality twice in the course of a full double circuit. The double loop is entirely contained within a single [[24-cell#Cell rings|cell ring]], where it follows chords connecting even (odd) vertices: typically opposite vertices of adjacent cells, two edge lengths apart.{{Efn|name=black and white}} Both &quot;halves&quot; of the double loop pass through each cell in the cell ring, but intersect only two even (odd) vertices in each even (odd) cell. Each pair of intersected vertices in an even (odd) cell lie opposite each other on the [[W:Möbius strip|Möbius strip]], exactly one edge length apart. Thus each cell has both helices passing through it, which are Clifford parallels{{Efn|name=Clifford parallels}} of opposite chirality at each pair of parallel points. Globally these two helices are a single connected circle of ''both'' chiralities, with no net [[W:Torsion of a curve|torsion]]. An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).{{Efn|name=one true circle}}|name=Clifford polygon}} Rather than a flat hexagon, it forms a [[W:Skew polygon|skew]] hexagram out of two three-sided 360 degree half-loops: open triangles joined end-to-end to each other in a six-sided Möbius loop.{{Efn|name=double threaded}}

Each 6-cell ring contains six such hexagram isoclines, three black and three white, that connect even and odd vertices respectively.{{Efn|Only one kind of 6-cell ring exists, not two different chiral kinds (right-handed and left-handed), because octahedra have opposing faces and form untwisted cell rings. In addition to two sets of three Clifford parallel{{Efn|name=Clifford parallels}} [[#Great hexagons|great hexagons]], three black and three white [[#Isoclinic rotations|isoclinic hexagram geodesics]] run through the [[#6-cell rings|6-cell ring]].{{Efn|name=hexagonal fibrations}} Each of these chiral skew [[W:Hexagram|hexagram]]s lies on a different kind of circle called an ''isocline'',{{Efn|name=not all isoclines are circles}} a helical circle [[W:Winding number|winding]] through all four dimensions instead of lying in a single plane.{{Efn|name=isoclinic geodesic}} These helical great circles occur in Clifford parallel [[W:Hopf fibration|fiber bundles]] just as ordinary planar great circles do. In the 6-cell ring, black and white hexagrams pass through even and odd vertices respectively, and miss the vertices in between, so the isoclines are disjoint.{{Efn|name=black and white}}|name=6-cell ring is not chiral}} Each of the three black-white pairs of isoclines belongs to one of the three fibrations in which the 6-cell ring occurs. Each fibration's right (or left) rotation traverses two black isoclines and two white isoclines in parallel, rotating all 24 vertices.{{Efn|name=missing the nearest vertices}}

Beginning at any vertex at one end of the column of six octahedra, we can follow an isoclinic path of {{radic|3}} chords of an isocline from octahedron to octahedron. In the 24-cell the {{radic|1}} edges are [[#Great hexagons|great hexagon]] edges (and octahedron edges); in the column of six octahedra we see six great hexagons running along the octahedra's edges. The {{radic|3}} chords are great hexagon diagonals, joining great hexagon vertices two {{radic|1}} edges apart. We find them in the ring of six octahedra running from a vertex in one octahedron to a vertex in the next octahedron, passing through the face shared by the two octahedra (but not touching any of the face's 3 vertices). Each {{radic|3}} chord is a chord of just one great hexagon (an edge of a [[#Great triangles|great triangle]] inscribed in that great hexagon), but successive {{radic|3}} chords belong to different great hexagons.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} At each vertex the isoclinic path of {{radic|3}} chords bends 60 degrees in two central planes{{Efn|Two central planes in which the path bends 60° at the vertex are (a) the great hexagon plane that the chord ''before'' the vertex belongs to, and (b) the great hexagon plane that the chord ''after'' the vertex belongs to. Plane (b) contains the 120° isocline chord joining the original vertex to a vertex in great hexagon plane (c), Clifford parallel to (a); the vertex moves over this chord to this next vertex. The angle of inclination between the Clifford parallel (isoclinic) great hexagon planes (a) and (c) is also 60°. In this 60° interval of the isoclinic rotation, great hexagon plane (a) rotates 60° within itself ''and'' tilts 60° in an orthogonal plane (not plane (b)) to become great hexagon plane (c). The three great hexagon planes (a), (b) and (c) are not orthogonal (they are inclined at 60° to each other), but (a) and (b) are two central hexagons in the same cuboctahedron, and (b) and (c) likewise in an orthogonal cuboctahedron.{{Efn|name=cuboctahedral hexagons}}}} at once: 60 degrees around the great hexagon that the chord before the vertex belongs to, and 60 degrees into the plane of a different great hexagon entirely, that the chord after the vertex belongs to.{{Efn|At each vertex there is only one adjacent great hexagon plane that the isocline can bend 60 degrees into: the isoclinic path is ''deterministic'' in the sense that it is linear, not branching, because each vertex in the cell ring is a place where just two of the six great hexagons contained in the cell ring cross. If each great hexagon is given edges and chords of a particular color (as in the 6-cell ring illustration), we can name each great hexagon by its color, and each kind of vertex by a hyphenated two-color name. The cell ring contains 18 vertices named by the 9 unique two-color combinations; each vertex and its antipodal vertex have the same two colors in their name, since when two great hexagons intersect they do so at antipodal vertices. Each isoclinic skew hexagram{{Efn|Each half of a skew hexagram is an open triangle of three {{radic|3}} chords, the two open ends of which are one {{radic|1}} edge length apart. The two halves, like the whole isocline, have no inherent chirality but the same parity-color (black or white). The halves are the two opposite &quot;edges&quot; of a [[W:Möbius strip|Möbius strip]] that is {{radic|1}} wide; it actually has only one edge, which is a single continuous circle with 6 chords.|name=skew hexagram}} contains one {{radic|3}} chord of each color, and visits 6 of the 9 different color-pairs of vertex.{{Efn|Each vertex of the 6-cell ring is intersected by two skew hexagrams of the same parity (black or white) belonging to different fibrations.{{Efn|name=6-cell ring is not chiral}}|name=hexagrams hitting vertex of 6-cell ring}} Each 6-cell ring contains six such isoclinic skew hexagrams, three black and three white.{{Efn|name=hexagrams missing vertex of 6-cell ring}}|name=Möbius double loop hexagram}} Thus the path follows one great hexagon from each octahedron to the next, but switches to another of the six great hexagons in the next link of the hexagram&lt;sub&gt;2&lt;/sub&gt; path. Followed along the column of six octahedra (and &quot;around the end&quot; where the column is bent into a ring) the path may at first appear to be zig-zagging between three adjacent parallel hexagonal central planes (like a [[W:Petrie polygon|Petrie polygon]]), but it is not: any isoclinic path we can pick out always zig-zags between ''two sets'' of three adjacent parallel hexagonal central planes, intersecting only every even (or odd) vertex and never changing its inherent even/odd parity, as it visits all six of the great hexagons in the 6-cell ring in rotation.{{Efn|The 24-cell's [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|dodecagon]] {12} and also (orthogonally) a skew [[W:Dodecagram|dodecagram]] {12/5} which zig-zags 90° left and right like the edges dividing the black and white squares on the [[W:Chessboard|chessboard]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell ''h&lt;sub&gt;1&lt;/sub&gt; is {12}, h&lt;sub&gt;2&lt;/sub&gt; is {12/5}''}} In contrast, the skew hexagram&lt;sub&gt;2&lt;/sub&gt; isocline does not zig-zag, and stays on one side or the other of the dividing line between black and white, like the [[W:Bishop (chess)|bishop]]s' paths along the diagonals of either the black or white squares of the chessboard.{{Efn|name=missing the nearest vertices}} The Petrie dodecagon is a circular helix of {{radic|1}} edges that zig-zag 90° left and right along 12 edges of 6 different octahedra (with 3 consecutive edges in each octahedron) in a 360° rotation. In contrast, the isoclinic hexagram&lt;sub&gt;2&lt;/sub&gt; has {{radic|3}} edges which all bend either left or right at every ''second'' vertex along a geodesic spiral of ''both'' chiralities (left and right){{Efn|name=Clifford polygon}} but only one color (black or white),{{Efn|name=black and white}} visiting one vertex of each of those same 6 octahedra in a 720° rotation.|name=Petrie dodecagram and Clifford hexagram}} When it has traversed one chord from each of the six great hexagons, after 720 degrees of isoclinic rotation (either left or right), it closes its skew hexagram and begins to repeat itself, circling again through the black (or white) vertices and cells.

At each vertex, there are four great hexagons{{Efn|Each pair of adjacent edges of a great hexagon has just one isocline curving alongside it,{{Efn|Each vertex of a 6-cell ring is missed by the two halves of the same Möbius double loop hexagram,{{Efn|name=Möbius double loop hexagram}} which curve past it on either side.|name=hexagrams missing vertex of 6-cell ring}} missing the vertex between the two edges (but not the way the {{radic|3}} edge of the great triangle inscribed in the great hexagon misses the vertex,{{Efn|The {{radic|3}} chord passes through the mid-edge of one of the 24-cell's {{radic|1}} radii. Since the 24-cell can be constructed, with its long radii, from {{radic|1}} triangles which meet at its center, this is a mid-edge of one of the six {{radic|1}} triangles in a great hexagon, as seen in the [[#Hypercubic chords|chord diagram]].|name=root 3 chord hits a mid-radius}} because the isocline is an arc on the surface not a chord). If we number the vertices around the hexagon 0-5, the hexagon has three pairs of adjacent edges connecting even vertices (one inscribed great triangle), and three pairs connecting odd vertices (the other inscribed great triangle). Even and odd pairs of edges have the arc of a black and a white isocline respectively curving alongside.{{Efn|name=black and white}} The three black and three white isoclines belong to the same 6-cell ring of the same fibration.{{Efn|name=Möbius double loop hexagram}}|name=isoclines at hexagons}} and four hexagram isoclines (all black or all white) that cross at the vertex.{{Efn|Each hexagram isocline hits only one end of an axis, unlike a great circle which hits both ends. Clifford parallel pairs of black and white isoclines from the same left-right pair of isoclinic rotations (the same fibration) do not intersect, but they hit opposite (antipodal) vertices of ''one'' of the 24-cell's 12 axes.|name=hexagram isoclines at an axis}} Four hexagram isoclines (two black and two white) comprise a unique (left or right) fiber bundle of isoclines covering all 24 vertices in each distinct (left or right) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of isoclines.{{Efn|The isoclines themselves are not left or right, only the bundles are. Each isocline is left ''and'' right.{{Efn|name=Clifford polygon}}}} There are 16 distinct hexagram isoclines in the 24-cell (8 black and 8 white).{{Efn|The 12 black-white pairs of hexagram isoclines in each fibration{{Efn|name=hexagram isoclines at an axis}} and the 16 distinct hexagram isoclines in the 24-cell form a [[W:Reye configuration|Reye configuration]] 12&lt;sub&gt;4&lt;/sub&gt;16&lt;sub&gt;3&lt;/sub&gt;, just the way the 24-cell's 12 axes and [[#Great hexagons|16 hexagons]] do. Each of the 12 black-white pairs occurs in one cell ring of each fibration of 4 hexagram isoclines, and each cell ring contains 3 black-white pairs of the 16 hexagram isoclines.|name=a right (left) isoclinic rotation is a Reye configuration}} Each isocline is a skew ''Clifford polygon'' of no inherent chirality, but acts as a left (or right) isocline when traversed by a left (or right) rotation in different fibrations.{{Efn|name=Clifford polygon}}

==== Helical octagrams and their isoclines ====
The 24-cell contains 18 helical [[W:Octagram|octagram]] isoclines (9 black and 9 white). Three pairs of octagram edge-helices are found in each of the three inscribed 16-cells, described elsewhere as the [[16-cell#Helical construction|helical construction of the 16-cell]]. In summary, each 16-cell can be decomposed (three different ways) into a left-right pair of 8-cell rings of {{radic|2}}-edged tetrahedral cells. Each 8-cell ring twists either left or right around an axial octagram helix of eight chords. In each 16-cell there are exactly 6 distinct helices, identical octagrams which each circle through all eight vertices. Each acts as either a left helix or a right helix or a Petrie polygon in each of the six distinct isoclinic rotations (three left and three right), and has no inherent chirality except in respect to a particular rotation. Adjacent vertices on the octagram isoclines are {{radic|2}} = 90° apart, so the circumference of the isocline is 4𝝅. An ''isoclinic'' rotation by 90° in great square invariant planes takes each vertex to its antipodal vertex, four vertices away in either direction along the isocline, and {{radic|4}} = 180° distant across the diameter of the isocline.

Each of the 3 fibrations of the 24-cell's 18 great squares corresponds to a distinct left (and right) isoclinic rotation in great square invariant planes. Each 60° step of the rotation takes 6 disjoint great squares (2 from each 16-cell) to great squares in a neighboring 16-cell, on [[16-cell#Helical construction|8-chord helical isoclines characteristic of the 16-cell]].{{Efn|As [[16-cell#Helical construction|in the 16-cell, the isocline is an octagram]] which intersects only 8 vertices, even though the 24-cell has more vertices closer together than the 16-cell. The isocline curve misses the additional vertices in between. As in the 16-cell, the first vertex it intersects is {{radic|2}} away. The 24-cell employs more octagram isoclines (3 in parallel in each rotation) than the 16-cell does (1 in each rotation). The 3 helical isoclines are Clifford parallel;{{Efn|name=Clifford parallels}} they spiral around each other in a triple helix, with the disjoint helices' corresponding vertex pairs joined by {{radic|1}} {{=}} 60° chords. The triple helix of 3 isoclines contains 24 disjoint {{radic|2}} edges (6 disjoint great squares) and 24 vertices, and constitutes a discrete fibration of the 24-cell, just as the 4-cell ring does.|name=octagram isoclines}}

In the 24-cell, these 18 helical octagram isoclines can be found within the six orthogonal [[#4-cell rings|4-cell rings]] of octahedra. Each 4-cell ring has cells bonded vertex-to-vertex around a great square axis, and we find antipodal vertices at opposite vertices of the great square. A {{radic|4}} chord (the diameter of the great square and of the isocline) connects them. [[#Boundary cells|Boundary cells]] describes how the {{radic|2}} axes of the 24-cell's octahedral cells are the edges of the 16-cell's tetrahedral cells, each tetrahedron is inscribed in a (tesseract) cube, and each octahedron is inscribed in a pair of cubes (from different tesseracts), bridging them.{{Efn|name=octahedral diameters}} The vertex-bonded octahedra of the 4-cell ring also lie in different tesseracts.{{Efn|Two tesseracts share only vertices, not any edges, faces, cubes (with inscribed tetrahedra), or octahedra (whose central square planes are square faces of cubes). An octahedron that touches another octahedron at a vertex (but not at an edge or a face) is touching an octahedron in another tesseract, and a pair of adjacent cubes in the other tesseract whose common square face the octahedron spans, and a tetrahedron inscribed in each of those cubes.|name=vertex-bonded octahedra}} The isocline's four {{radic|4}} diameter chords form an [[W:Octagram#Star polygon compounds|octagram&lt;sub&gt;8{4}=4{2}&lt;/sub&gt;]] with {{radic|4}} edges that each run from the vertex of one cube and octahedron and tetrahedron, to the vertex of another cube and octahedron and tetrahedron (in a different tesseract), straight through the center of the 24-cell on one of the 12 {{radic|4}} axes.

The octahedra in the 4-cell rings are vertex-bonded to more than two other octahedra, because three 4-cell rings (and their three axial great squares, which belong to different 16-cells) cross at 90° at each bonding vertex. At that vertex the octagram makes two right-angled turns at once: 90° around the great square, and 90° orthogonally into a different 4-cell ring entirely. The 180° four-edge arc joining two ends of each {{radic|4}} diameter chord of the octagram runs through the volumes and opposite vertices of two face-bonded {{radic|2}} tetrahedra (in the same 16-cell), which are also the opposite vertices of two vertex-bonded octahedra in different 4-cell rings (and different tesseracts). The [[W:Octagram|720° octagram]] isocline runs through 8 vertices of the four-cell ring and through the volumes of 16 tetrahedra. At each vertex, there are three great squares and six octagram isoclines (three black-white pairs) that cross at the vertex.{{Efn|name=completely orthogonal Clifford parallels are special}}

This is the characteristic rotation of the 16-cell, ''not'' the 24-cell's characteristic rotation, and it does not take whole 16-cells ''of the 24-cell'' to each other the way the [[#Helical hexagrams and their isoclines|24-cell's rotation in great hexagon planes]] does.{{Efn|The [[600-cell#Squares and 4𝝅 octagrams|600-cell's isoclinic rotation in great square planes]] takes whole 16-cells to other 16-cells in different 24-cells.}}

{| class=&quot;wikitable&quot; width=610
!colspan=5|Five ways of looking at a [[W:Skew polygon|skew]] [[W:24-gon#Related polygons|24-gram]]
|-
![[16-cell#Rotations|Edge path]]
![[W:Petrie polygon|Petrie polygon]]s
![[600-cell#Squares and 4𝝅 octagrams|In a 600-cell]]
![[#Great squares|Discrete fibration]]
![[16-cell#Helical construction|Diameter chords]]
|-
![[16-cell#Helical construction|16-cells]]&lt;sub&gt;3{3/8}&lt;/sub&gt;
![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]&lt;sub&gt;2{12}&lt;/sub&gt;
![[W:24-gon#Related polygons|24-gram]]&lt;sub&gt;{24/5}&lt;/sub&gt;
![[#Great squares|Squares]]&lt;sub&gt;6{4}&lt;/sub&gt;
![[W:24-gon#Related polygons|&lt;sub&gt;{24/12}={12/2}&lt;/sub&gt;]]
|-
|align=center|[[File:Regular_star_figure_3(8,3).svg|120px]]
|align=center|[[File:Regular_star_figure_2(12,1).svg|120px]]
|align=center|[[File:Regular_star_polygon_24-5.svg|120px]]
|align=center|[[File:Regular_star_figure_6(4,1).svg|120px]]
|align=center|[[File:Regular_star_figure_12(2,1).svg|120px]]
|-
|The 24-cell's three inscribed Clifford parallel 16-cells revealed as disjoint 8-point 4-polytopes with {{radic|2}} edges.{{Efn|name=octagram isoclines}}
|2 [[W:Skew polygon|skew polygon]]s of 12 {{radic|1}} edges each. The 24-cell can be decomposed into 2 disjoint zig-zag [[W:Dodecagon|dodecagon]]s (4 different ways).{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon ''h&lt;sub&gt;1&lt;/sub&gt;'' is {12} }}
|In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length &lt;big&gt;φ&lt;/big&gt; {{=}} {{radic|2.𝚽}} connect all 24-cells in [[600-cell#Squares and 4𝝅 octagrams|24-chord circuits]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); 24-cell Petrie polygon orthogonal ''h&lt;sub&gt;2&lt;/sub&gt;'' is [[W:Dodecagon#Related figures|{12/5}]], half of [[W:24-gon#Related polygons|{24/5}]] as each Petrie polygon is half the 24-cell}}
|Their isoclinic rotation takes 6 Clifford parallel (disjoint) great squares with {{radic|2}} edges to each other.
|Two vertices four {{radic|2}} chords apart on the circular isocline are antipodal vertices joined by a {{radic|4}} axis.
|}

===Characteristic orthoscheme===
{| class=&quot;wikitable floatright&quot;
!colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); &quot;24-cell&quot;}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); &quot;dihedral angles&quot;}}
|-
!align=right|𝒍
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;120°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{2\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|𝟀
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}} \approx 0.577&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[W:Golden ratio|golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|𝟁
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}} \approx 0.289&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;60°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{3}&lt;/math&gt;&lt;/small&gt;
|-
|
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|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;45°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{4}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}} = 0.5&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^3/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}} \approx 0.408&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;30°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{6}&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;90°&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\tfrac{\pi}{2}&lt;/math&gt;&lt;/small&gt;
|-
|
|
|
|
|
|-
!align=right|&lt;small&gt;&lt;math&gt;_0R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_1R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}} \approx 0.866&lt;/math&gt;&lt;/small&gt;{{Efn|name=root 3/4}}
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_2R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}} \approx 0.816&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|&lt;small&gt;&lt;math&gt;_3R^4/l&lt;/math&gt;&lt;/small&gt;
|align=center|&lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}} \approx 0.707&lt;/math&gt;&lt;/small&gt;
|align=center|
|align=center|
|align=center|
|align=center|
|}
Every regular 4-polytope has its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic 4-orthoscheme]], an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{Efn|name=characteristic orthoscheme}} The '''characteristic 5-cell of the regular 24-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, which can be read as a list of the dihedral angles between its mirror facets.{{Efn|For a regular ''k''-polytope, the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] of the characteristic ''k-''orthoscheme is the ''k''-polytope's diagram without the [[W:Coxeter-Dynkin diagram#Application with uniform polytopes|generating point ring]]. The regular ''k-''polytope is subdivided by its symmetry (''k''-1)-elements into ''g'' instances of its characteristic ''k''-orthoscheme that surround its center, where ''g'' is the ''order'' of the ''k''-polytope's [[W:Coxeter group|symmetry group]].{{Sfn|Coxeter|1973|pp=130-133|loc=§7.6 The symmetry group of the general regular polytope}}}} It is an irregular [[W:Hyperpyramid|tetrahedral pyramid]] based on the [[W:Octahedron#Characteristic orthoscheme|characteristic tetrahedron of the regular octahedron]]. The regular 24-cell is subdivided by its symmetry hyperplanes into 1152 instances of its characteristic 5-cell that all meet at its center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}}

The characteristic 5-cell (4-orthoscheme) has four more edges than its base characteristic tetrahedron (3-orthoscheme), joining the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 24-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of the regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. The five vertices of the 4-orthoscheme always include one regular 4-polytope vertex, one regular 4-polytope edge center, one regular 4-polytope face center, one regular 4-polytope cell center, and the regular 4-polytope center. Those five vertices (in that order) comprise a path along four mutually perpendicular edges (that makes three right angle turns), the characteristic feature of a 4-orthoscheme. The 4-orthoscheme has five dissimilar 3-orthoscheme facets.|name=characteristic radii}} If the regular 24-cell has radius and edge length 𝒍 = 1, its characteristic 5-cell's ten edges have lengths &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt; around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt; (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus &lt;small&gt;&lt;math&gt;1&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{3}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{2}{3}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt; (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{4}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{12}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{6}}&lt;/math&gt;&lt;/small&gt;, &lt;small&gt;&lt;math&gt;\sqrt{\tfrac{1}{2}}&lt;/math&gt;&lt;/small&gt;, first from a 24-cell vertex to a 24-cell edge center, then turning 90° to a 24-cell face center, then turning 90° to a 24-cell octahedral cell center, then turning 90° to the 24-cell center.

=== Reflections ===
The 24-cell can be [[#Tetrahedral constructions|constructed by the reflections of its characteristic 5-cell]] in its own facets (its tetrahedral mirror walls).{{Efn|The reflecting surface of a (3-dimensional) polyhedron consists of 2-dimensional faces; the reflecting surface of a (4-dimensional) [[W:Polychoron|polychoron]] consists of 3-dimensional cells.}} Reflections and rotations are related: a reflection in an ''even'' number of ''intersecting'' mirrors is a rotation.{{Sfn|Coxeter|1973|pp=33-38|loc=§3.1 Congruent transformations}} Consequently, regular polytopes can be generated by reflections or by rotations. For example, any [[#Isoclinic rotations|720° isoclinic rotation]] of the 24-cell in a hexagonal invariant plane takes ''each'' of the 24 vertices to and through 5 other vertices and back to itself, on a skew [[#Helical hexagrams and their isoclines|hexagram&lt;sub&gt;2&lt;/sub&gt; geodesic isocline]] that winds twice around the 3-sphere on every ''second'' vertex of the hexagram. Any set of [[#The 3 Cartesian bases of the 24-cell|four orthogonal pairs of antipodal vertices]] (the 8 vertices of one of the [[#Relationships among interior polytopes|three inscribed 16-cells]]) performing ''half'' such an orbit visits 3 * 8 = 24 distinct vertices and [[#Clifford parallel polytopes|generates the 24-cell]] sequentially in 3 steps of a single 360° isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 24 vertices simultaneously by reflection.

Tracing the orbit of ''one'' such 16-cell vertex during the 360° isoclinic rotation reveals more about the relationship between reflections and rotations as generative operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} The vertex follows an [[#Helical hexagrams and their isoclines|isocline]] (a doubly curved geodesic circle) rather than an ordinary great circle.{{Efn|name=360 degree geodesic path visiting 3 hexagonal planes}} The isocline connects vertices two edge lengths apart, but curves away from the great circle path over the two edges connecting those vertices, missing the vertex in between.{{Efn|name=isocline misses vertex}} Although the isocline does not follow any one great circle, it is contained within a ring of another kind: in the 24-cell it stays within a [[#6-cell rings|6-cell ring]] of spherical{{Sfn|Coxeter|1973|p=138|ps=; &quot;We allow the Schläfli symbol {p,..., v} to have three different meanings: a Euclidean polytope, a spherical polytope, and a spherical honeycomb. This need not cause any confusion, so long as the situation is frankly recognized. The differences are clearly seen in the concept of dihedral angle.&quot;}} octahedral cells, intersecting one vertex in each cell, and passing through the volume of two adjacent cells near the missed vertex.

=== Chiral symmetry operations ===

A [[W:Symmetry operation|symmetry operation]] is a rotation or reflection which leaves the object indistinguishable from itself before the transformation. The 24-cell has 1152 distinct symmetry operations (576 rotations and 576 reflections). Each rotation is equivalent to two [[#Reflections|reflections]], in a distinct pair of non-parallel mirror planes.{{Efn|name=transformations}}

Pictured are sets of disjoint [[#Geodesics|great circle polygons]], each in a distinct central plane of the 24-cell. For example, {24/4}=4{6} is an orthogonal projection of the 24-cell picturing 4 of its [16] great hexagon planes.{{Efn|name=four hexagonal fibrations}} The 4 planes lie Clifford parallel to the projection plane and to each other, and their great polygons collectively constitute a discrete [[W:Hopf fibration|Hopf fibration]] of 4 non-intersecting great circles which visit all 24 vertices just once.

Each row of the table describes a class of distinct rotations. Each '''rotation class''' takes the '''left planes''' pictured to the corresponding '''right planes''' pictured.{{Efn|The left planes are Clifford parallel, and the right planes are Clifford parallel; each set of planes is a fibration. Each left plane is Clifford parallel to its corresponding right plane in an isoclinic rotation,{{Efn|In an ''isoclinic'' rotation each invariant plane is Clifford parallel to the plane it moves to, and they do not intersect at any time (except at the central point). In a ''simple'' rotation the invariant plane intersects the plane it moves to in a line, and moves to it by rotating around that line.|name=plane movement in rotations}} but the two sets of planes are not all mutually Clifford parallel; they are different fibrations, except in table rows where the left and right planes are the same set.}} The vertices of the moving planes move in parallel along the polygonal '''isocline''' paths pictured. For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation class consists of [32] distinct rotational displacements by an arc-distance of {{sfrac|2𝝅|3}} = 120° between 16 great hexagon planes represented by quaternion group &lt;math&gt;q7&lt;/math&gt; and a corresponding set of 16 great hexagon planes represented by quaternion group &lt;math&gt;q8&lt;/math&gt;.{{Efn|A quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; corresponds to a distinct set of Clifford parallel great circle polygons, e.g. &lt;math&gt;q7&lt;/math&gt; corresponds to a set of four disjoint great hexagons.{{Efn|[[File:Regular_star_figure_4(6,1).svg|thumb|200px|The 24-cell as a compound of four non-intersecting great hexagons {24/4}=4{6}.]]There are 4 sets of 4 disjoint great hexagons in the 24-cell (of a total of [16] distinct great hexagons), designated &lt;math&gt;q7&lt;/math&gt;, &lt;math&gt;-q7&lt;/math&gt;, &lt;math&gt;q8&lt;/math&gt; and &lt;math&gt;-q8&lt;/math&gt;.{{Efn|name=union of q7 and q8}} Each named set of 4 Clifford parallel{{Efn|name=Clifford parallels}} hexagons comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=four hexagonal fibrations}} Note that &lt;math&gt;q_n&lt;/math&gt; and &lt;math&gt;-{q_n}&lt;/math&gt; generally are distinct sets. The corresponding vertices of the &lt;math&gt;q_n&lt;/math&gt; planes and the &lt;math&gt;-{q_n}&lt;/math&gt; planes are 180° apart.{{Efn|name=two angles between central planes}}|name=quaternion group}} One of the [32] distinct rotations of this class moves the representative [[#Great hexagons|vertex coordinate]] &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; to the vertex coordinate &lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;.{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in standard (vertex-up) orientation is &lt;math&gt;(0,0,1,0)&lt;/math&gt;, the Cartesian &quot;north pole&quot;. Thus e.g. &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; designates a {{radic|1}} chord of 60° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great hexagons|great hexagon]], intersecting the north and south poles. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the north and south poles. This quaternion coordinate &lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt; is thus representative of the 4 disjoint great hexagons pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [16] great hexagons (four fibrations of great hexagons) that occur in the 24-cell.{{Efn|name=four hexagonal fibrations}}|name=north pole relative coordinate}}

{| class=wikitable style=&quot;white-space:nowrap;text-align:center&quot;
!colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F&lt;sub&gt;4&lt;/sub&gt;'']] {{Sfn|Mamone|Pileio|Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439}}
|-
!Isocline{{Efn|An ''isocline'' is the circular geodesic path taken by a vertex that lies in an invariant plane of rotation, during a complete revolution. In an [[#Isoclinic rotations|isoclinic rotation]] every vertex lies in an invariant plane of rotation, and the isocline it rotates on is a helical geodesic circle that winds through all four dimensions, not a simple geodesic great circle in the plane. In a [[#Simple rotations|simple rotation]] there is only one invariant plane of rotation, and each vertex that lies in it rotates on a simple geodesic great circle in the plane. Both the helical geodesic isocline of an isoclinic rotation and the simple geodesic isocline of a simple rotation are great circles, but to avoid confusion between them we generally reserve the term ''isocline'' for the former, and reserve the term ''great circle'' for the latter, an ordinary great circle in the plane. Strictly, however, the latter is an isocline of circumference &lt;math&gt;2\pi r&lt;/math&gt;, and the former is an isocline of circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;.{{Efn|name=isoclinic geodesic}}|name=isocline}}
!colspan=4|Rotation class{{Efn|Each class of rotational displacements (each table row) corresponds to a distinct rigid left (and right) [[#Isoclinic rotations|isoclinic rotation]] in multiple invariant planes concurrently.{{Efn|name=invariant planes of an isoclinic rotation}} The '''Isocline''' is the path followed by a vertex,{{Efn|name=isocline}} which is a helical geodesic circle that does not lie in any one central plane. Each rotational displacement takes one invariant '''Left plane''' to the corresponding invariant '''Right plane''', with all the left (or right) displacements taking place concurrently.{{Efn|name=plane movement in rotations}} Each left plane is separated from the corresponding right plane by two equal angles,{{Efn|name=two angles between central planes}} each equal to one half of the arc-angle by which each vertex is displaced (the angle and distance that appears in the '''Rotation class''' column).|name=isoclinic rotation}}
!colspan=5|Left planes &lt;math&gt;ql&lt;/math&gt;{{Efn|In an [[#Isoclinic rotations|isoclinic rotation]], all the '''Left planes''' move together, remain Clifford parallel while moving, and carry all their points with them to the '''Right planes''' as they move: they are invariant planes.{{Efn|name=plane movement in rotations}} Because the left (and right) set of central polygons are a fibration covering all the vertices, every vertex is a point carried along in an invariant plane.|name=invariant planes of an isoclinic rotation}}
!colspan=5|Right planes &lt;math&gt;qr&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. Each disjoint triangle can be seen as a skew {6/2} [[W:Hexagram|hexagram]] with {{radic|3}} edges: two open skew triangles with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The hexagram projects to a single triangle in two dimensions because it skews through all four dimensions. Those 4 disjoint skew [[#Helical hexagrams and their isoclines|hexagram isoclines]] are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 60° like wheels ''and'' 60° orthogonally like coins flipping, displacing each vertex by 120°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only three skew hexagram isoclines, not six, because opposite vertices of each hexagon ride on opposing rails of the same Clifford hexagram, in the same (not opposite) rotational direction.{{Efn|name=Clifford polygon}}}} Alternatively, the 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 4 hexagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[32]R_{q7,q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]{{Efn|name=four hexagonal fibrations}}&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;{{Efn|name=north pole relative coordinate}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/4}=4{3} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|3}} chords. The 4 triangles can be seen as 8 disjoint triangles: 4 pairs of Clifford parallel [[#Great triangles|great triangles]], where two opposing great triangles lie in the same [[#Great hexagons|great hexagon central plane]], so a fibration of 4 Clifford parallel great hexagon planes is represented, as in the 4 left planes of this rotation class (table row).{{Efn|name=four hexagonal fibrations}}|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/2}=2{6} dodecagram]], each point represents two vertices, and each line represents multiple 24-cell edges. Each disjoint hexagon can be seen as a skew {12} [[W:Dodecagon|dodecagon]], a Petrie polygon of the 24-cell, by viewing it as two open skew hexagons with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅. The dodecagon projects to a single hexagon in two dimensions because it skews through all four dimensions. Those 2 disjoint skew dodecagons are the Clifford parallel circular vertex paths of the fibration's characteristic left (and right) [[#Isoclinic rotations|isoclinic rotation]].{{Efn|name=isoclinic geodesic}} The 4 Clifford parallel great hexagons of the fibration are invariant planes of this rotation. The great hexagons rotate in incremental displacements of 30° like wheels ''and'' 30° orthogonally like coins flipping, displacing each vertex by 60°, as their vertices move along parallel helical isocline paths through successive Clifford parallel hexagon planes.{{Efn|Each hexagon rides on only two parallel dodecagon isoclines, not six, because only alternate vertices of each hexagon ride on different dodecagon rails; the three vertices of each great triangle inscribed in the great hexagon occupy the same dodecagon Petrie polygon, four vertices apart, and they circulate on that isocline.{{Efn|name=Clifford polygon}}}} Alternatively, the 2 hexagons can be seen as 4 disjoint hexagons: 2 pairs of Clifford parallel great hexagons, so a fibration of 4 Clifford parallel great hexagon planes is represented.{{Efn|name=four hexagonal fibrations}} This illustrates that the 2 dodecagon isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 4 great hexagons.|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q8}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {12}
|colspan=4|&lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q8}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (30° away) it passes directly over the mid-point of a 24-cell edge.}} Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q8}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q8}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {1}
|colspan=4|&lt;math&gt;[32]R_{q7,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left hexagon rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q7}&lt;/math&gt;&lt;br&gt;[16] 4𝝅 {2}
|colspan=4|&lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[32]R_{q7,-q7}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex three vertices away (180° {{=}} {{radic|4}} away),{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left hexagon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right hexagon plane. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q7}&lt;/math&gt;&lt;br&gt;[16] 2𝝅 {6}
|colspan=4|&lt;math&gt;(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {12}
|colspan=4|&lt;math&gt;[16]R_{q7,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,q1}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left hexagon rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|This ''hybrid isoclinic rotation'' carries the two kinds of [[#Geodesics|central planes]] to each other: great square planes [[16-cell#Coordinates|characteristic of the 16-cell]] and great hexagon (great triangle) planes [[#Great hexagons|characteristic of the 24-cell]].{{Efn|The edges and 4𝝅 characteristic [[16-cell#Rotations|rotations of the 16-cell]] lie in the great square central planes. Rotations of this type are an expression of the [[W:Hyperoctahedral group|&lt;math&gt;B_4&lt;/math&gt; symmetry group]]. The edges and 4𝝅 characteristic [[#Rotations|rotations of the 24-cell]] lie in the great hexagon (great triangle) central planes. Rotations of this type are an expression of the [[W:F4 (mathematics)|&lt;math&gt;F_4&lt;/math&gt; symmetry group]].|name=edge rotation planes}} This is possible because some great hexagon planes lie Clifford parallel to some great square planes.{{Efn|Two great circle polygons either intersect in a common axis, or they are Clifford parallel (isoclinic) and share no vertices.{{Efn||name=two angles between central planes}} Three great squares and four great hexagons intersect at each 24-cell vertex. Each great hexagon intersects 9 distinct great squares, 3 in each of its 3 axes, and lies Clifford parallel to the other 9 great squares. Each great square intersects 8 distinct great hexagons, 4 in each of its 2 axes, and lies Clifford parallel to the other 8 great hexagons.|name=hybrid isoclinic planes}}|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]{{Efn|[[File:Regular_star_figure_6(4,1).svg|thumb|200px|The 24-cell as a compound of six non-intersecting great squares {24/6}=6{4}.]]There are 3 sets of 6 disjoint great squares in the 24-cell (of a total of [18] distinct great squares),{{Efn|The 24-cell has 18 great squares, in 3 disjoint sets of 6 mutually orthogonal great squares comprising a 16-cell.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Within each 16-cell are 3 sets of 2 completely orthogonal great squares, so each great square is disjoint not only from all the great squares in the other two 16-cells, but also from one other great square in the same 16-cell. Each great square is disjoint from 13 others, and shares two vertices (an axis) with 4 others (in the same 16-cell).|name=unions of q1 q2 q3}} designated &lt;math&gt;\pm q1&lt;/math&gt;, &lt;math&gt;\pm q2&lt;/math&gt;, and &lt;math&gt;\pm q3&lt;/math&gt;. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups &lt;math&gt;q1&lt;/math&gt; and &lt;math&gt;-{q1}&lt;/math&gt; (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares &lt;math&gt;\pm q1&lt;/math&gt; has two names, used in the left (or right) rotational context, because it constitutes both a left and a right fibration of great squares.|name=two quaternion group names for square fibrations}} of 6 Clifford parallel{{Efn|name=Clifford parallels}} squares comprises a [[#Chiral symmetry operations|discrete fibration]] covering all 24 vertices.|name=three square fibrations}}&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}&lt;br&gt;[[File:Regular_star_figure_4(3,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7,-q1}&lt;/math&gt;&lt;br&gt;[8] 4𝝅 {6/2}
|colspan=4|&lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[16]R_{q7,-q1}&lt;/math&gt; isoclinic rotation in hexagon invariant planes takes each vertex to a vertex two vertices away (120° {{=}} {{radic|3}} away), without passing through any intervening vertices. Each left hexagon rotates 60° (like a wheel) at the same time that it tilts sideways by 60° (in an orthogonal central plane) into its corresponding right square plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 6 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq7,-q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[8] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|2𝝅|3}}
|120°
|{{radic|3}}
|1.732~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {1}
|colspan=4|&lt;math&gt;[36]R_{q6,q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;{{Efn|The representative coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is not a vertex of the unit-radius 24-cell in standard (vertex-up) orientation, it is the center of an octahedral cell. Some of the 24-cell's lines of symmetry (Coxeter's &quot;reflecting circles&quot;) run through cell centers rather than through vertices, and quaternion group &lt;math&gt;q6&lt;/math&gt; corresponds to a set of those. However, &lt;math&gt;q6&lt;/math&gt; also corresponds to the set of great squares pictured, which lie orthogonal to those cells (completely disjoint from the cell).{{Efn|A quaternion Cartesian coordinate designates a vertex joined to a ''top vertex'' by one instance of a [[#Hypercubic chords|distinct chord]]. The conventional top vertex of a [[#Great hexagons|unit radius 4-polytope]] in ''cell-first'' orientation is &lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;. Thus e.g. &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; designates a {{radic|2}} chord of 90° arc-length. Each such distinct chord is an edge of a distinct [[#Geodesics|great circle polygon]], in this example a [[#Great squares|great square]], intersecting the top vertex. Great circle polygons occur in sets of Clifford parallel central planes, each set of disjoint great circles comprising a discrete [[W:Hopf fibration|Hopf fibration]] that intersects every vertex just once. One great circle polygon in each set intersects the top vertex. This quaternion coordinate &lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt; is thus representative of the 6 disjoint great squares pictured, a quaternion group{{Efn|name=quaternion group}} which comprise one distinct fibration of the [18] great squares (three fibrations of great squares) that occur in the 24-cell.{{Efn|name=three square fibrations}}|name=north cell relative coordinate}}|name=lines of symmetry}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q6}&lt;/math&gt;&lt;br&gt;[18] 4𝝅 {2}
|colspan=4|&lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt;{{Efn|The &lt;math&gt;[36]R_{q6,-q6}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q6}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q6}&lt;/math&gt;&lt;br&gt;[18] 2𝝅 {4}
|colspan=4|&lt;math&gt;(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/9}=3{8/3}]]{{Efn|In this orthogonal projection of the 24-point 24-cell to a [[W:Dodecagon#Related figures|{12/3}{{=}}3{4} dodecagram]], each point represents two vertices, and each line represents multiple {{radic|2}} chords. Each disjoint square can be seen as a skew {8/3} [[W:Octagram|octagram]] with {{radic|2}} edges: two open skew squares with their opposite ends connected in a [[W:Möbius strip|Möbius loop]] with a circumference of 4𝝅, visible in the {24/9}{{=}}3{8/3} orthogonal projection.{{Efn|[[File:Regular_star_figure_3(8,3).svg|thumb|200px|Icositetragon {24/9}{{=}}3{8/3} is a compound of three octagrams {8/3}, as the 24-cell is a compound of three 16-cells.]]This orthogonal projection of a 24-cell to a 24-gram {24/9}{{=}}3{8/3} exhibits 3 disjoint [[16-cell#Helical construction|octagram {8/3} isoclines of a 16-cell]], each of which is a circular isocline path through the 8 vertices of one of the 3 disjoint 16-cells inscribed in the 24-cell.}} The octagram projects to a single square in two dimensions because it skews through all four dimensions. Those 3 disjoint [[16-cell#Helical construction|skew octagram isoclines]] are the circular vertex paths characteristic of an [[#Helical octagrams and their isoclines|isoclinic rotation in great square planes]], in which the 6 Clifford parallel great squares are invariant rotation planes. The great squares rotate 90° like wheels ''and'' 90° orthogonally like coins flipping, displacing each vertex by 180°, so each vertex exchanges places with its antipodal vertex. Each octagram isocline circles through the 8 vertices of a disjoint 16-cell. Alternatively, the 3 squares can be seen as a fibration of 6 Clifford parallel squares.{{Efn|name=three square fibrations}} This illustrates that the 3 octagram isoclines also correspond to a distinct fibration, in fact the ''same'' fibration as 6 squares.|name=octagram}}&lt;br&gt;[[File:Regular_star_figure_3(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6,-q4}&lt;/math&gt;&lt;br&gt;[72] 4𝝅 {8/3}
|colspan=4|&lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[144]R_{q6,-q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left square rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right square plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq6,-q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q6}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q4}&lt;/math&gt;&lt;br&gt;[72] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|𝝅
|180°
|{{radic|4}}
|2
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q4,q4}&lt;/math&gt;&lt;br&gt;[36] 4𝝅 {1}
|colspan=4|&lt;math&gt;[72]R_{q4,q4}&lt;/math&gt;{{Efn|The &lt;math&gt;[72]R_{q4,q4}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex through a 360° rotation and back to itself (360° {{=}} {{radic|0}} away), without passing through any intervening vertices. Each left square rotates 180° (like a wheel) at the same time that it tilts sideways by 180° (in an orthogonal central plane) into its corresponding right square plane. Repeated 2 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq4,q4}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q4}&lt;/math&gt;&lt;br&gt;[36] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|2𝝅
|360°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: #E6FFEE;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}&lt;br&gt;[[File:Regular_star_figure_2(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q7}&lt;/math&gt;&lt;br&gt;[48] 4𝝅 {12}
|colspan=4|&lt;math&gt;[96]R_{q2,q7}&lt;/math&gt;{{Efn|The &lt;math&gt;[96]R_{q2,q7}&lt;/math&gt; isoclinic rotation in great hexagon invariant planes takes each vertex to a vertex one vertex away (60° {{=}} {{radic|1}} away), without passing through any intervening vertices. Each left square rotates 30° (like a wheel) at the same time that it tilts sideways by 30° (in an orthogonal central plane) into its corresponding right hexagon plane.{{Efn|name=hybrid isoclinic rotation}} Repeated 12 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q7}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]&lt;br&gt;[[File:Regular_star_figure_4(6,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q7}&lt;/math&gt;&lt;br&gt;[48] 2𝝅 {6}
|colspan=4|&lt;math&gt;(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})&lt;/math&gt;
|- style=&quot;background: #E6FFEE;&quot;|
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|3}}
|60°
|{{radic|1}}
|1
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,-q2}&lt;/math&gt;&lt;br&gt;[9] 4𝝅 {2}
|colspan=4|&lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt;{{Efn|The &lt;math&gt;[18]R_{q2,-q2}&lt;/math&gt; isoclinic rotation in great square invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left square rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right square plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,-q2}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]&lt;br&gt;[[File:Regular_star_figure_6(4,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q2}&lt;/math&gt;&lt;br&gt;[9] 2𝝅 {4}
|colspan=4|&lt;math&gt;(0,0,0,-1)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2,q1}&lt;/math&gt;&lt;br&gt;[12] 4𝝅 {2}
|colspan=4|&lt;math&gt;[12]R_{q2,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[12]R_{q2,q1}&lt;/math&gt; isoclinic rotation in great digon invariant planes takes each vertex to a vertex 90° {{=}} {{radic|2}} away, without passing through any intervening vertices.{{Efn|At the mid-point of the isocline arc (45° away) it passes directly over the mid-point of a 24-cell edge.}} Each left digon rotates 45° (like a wheel) at the same time that it tilts sideways by 45° (in an orthogonal central plane) into its corresponding right digon plane. Repeated 8 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq2,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q2}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(0,0,0,1)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]&lt;br&gt;[[File:Regular_polygon_24.svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,q1}&lt;/math&gt;&lt;br&gt;[0] 0𝝅 {1}
|colspan=4|&lt;math&gt;[1]R_{q1,q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,q1}&lt;/math&gt; rotation is the ''identity operation'' of the 24-cell, in which no points move.|name=Rq1,q1}}
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[0] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|0
|0°
|{{radic|0}}
|0
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|- style=&quot;background: white;&quot;|
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1,-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt;{{Efn|The &lt;math&gt;[1]R_{q1,-q1}&lt;/math&gt; rotation is the ''central inversion'' of the 24-cell. This isoclinic rotation in great digon invariant planes takes each vertex to a vertex 180° {{=}} {{radic|4}} away,{{Efn|name=quaternion group}} without passing through any intervening vertices. Each left digon rotates 90° (like a wheel) at the same time that it tilts sideways by 90° (in an orthogonal central plane) into its corresponding right digon plane, ''which in this rotation is the completely orthogonal plane''. Repeated 4 times, this rotational displacement turns the 24-cell through 720° and returns it to its original orientation.|name=Rq1,-q1}} 
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(1,0,0,0)&lt;/math&gt;
|rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]&lt;br&gt;[[File:Regular_star_figure_12(2,1).svg|100px]]&lt;br&gt;&lt;math&gt;^{-q1}&lt;/math&gt;&lt;br&gt;[12] 2𝝅 {2}
|colspan=4|&lt;math&gt;(-1,0,0,0)&lt;/math&gt;
|- style=&quot;background: white;&quot;|
|𝝅
|180°
|{{radic|4}}
|2
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|{{sfrac|𝝅|2}}
|90°
|{{radic|2}}
|1.414~
|}

In a rotation class &lt;math&gt;[d]{R_{ql,qr}}&lt;/math&gt; each quaternion group &lt;math&gt;\pm{q_n}&lt;/math&gt; may be representative not only of its own fibration of Clifford parallel planes{{Efn|name=quaternion group}} but also of the other congruent fibrations.{{Efn|name=four hexagonal fibrations}} For example, rotation class &lt;math&gt;[4]R_{q7,q8}&lt;/math&gt; takes the 4 hexagon planes of &lt;math&gt;q7&lt;/math&gt; to the 4 hexagon planes of &lt;math&gt;q8&lt;/math&gt; which are 120° away, in an isoclinic rotation. But in a rigid rotation of this kind,{{Efn|name=invariant planes of an isoclinic rotation}} all [16] hexagon planes move in congruent rotational displacements, so this rotation class also includes &lt;math&gt;[4]R_{-q7,-q8}&lt;/math&gt;, &lt;math&gt;[4]R_{q8,q7}&lt;/math&gt; and &lt;math&gt;[4]R_{-q8,-q7}&lt;/math&gt;. The name &lt;math&gt;[16]R_{q7,q8}&lt;/math&gt; is the conventional representation for all [16] congruent plane displacements.

These rotation classes are all subclasses of &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; which has [32] distinct rotational displacements rather than [16] because there are two [[W:Chiral|chiral]] ways to perform any class of rotations, designated its ''left rotations'' and its ''right rotations''. The [16] left displacements of this class are not congruent with the [16] right displacements, but enantiomorphous like a pair of shoes.{{Efn|A ''right rotation'' is performed by rotating the left and right planes in the &quot;same&quot; direction, and a ''left rotation'' is performed by rotating left and right planes in &quot;opposite&quot; directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is &quot;up&quot; on each of the 4 coordinate axes. Left and right rotations are [[chiral]] enantiomorphous ''shapes'' (like a pair of shoes), not opposite rotational ''directions''. Both left and right rotations can be performed in either the positive or negative rotational direction (from left planes to right planes, or right planes to left planes), but that is an additional distinction.{{Efn|name=clasped hands}}|name=chirality versus direction}} Each left (or right) isoclinic rotation takes [16] left planes to [16] right planes, but the left and right planes correspond differently in the left and right rotations. The left and right rotational displacements of the same left plane take it to different right planes.

Each rotation class (table row) describes a distinct left (and right) [[#Isoclinic rotations|isoclinic rotation]]. The left (or right) rotations carry the left planes to the right planes simultaneously,{{Efn|name=plane movement in rotations}} through a characteristic rotation angle.{{Efn|name=two angles between central planes}} For example, the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; rotation moves all [16] hexagonal planes at once by {{sfrac|2𝝅|3}} = 120° each. Repeated 6 times, this left (or right) isoclinic rotation moves each plane 720° and back to itself in the same [[W:Orientation entanglement|orientation]], passing through all 4 planes of the &lt;math&gt;q7&lt;/math&gt; left set and all 4 planes of the &lt;math&gt;q8&lt;/math&gt; right set once each.{{Efn|The &lt;math&gt;\pm q7&lt;/math&gt; and &lt;math&gt;\pm q8&lt;/math&gt; sets of planes are not disjoint; the union of any two of these four sets is a set of 6 planes. The left (versus right) isoclinic rotation of each of these rotation classes (table rows) visits a distinct left (versus right) circular sequence of the same set of 6 Clifford parallel planes.|name=union of q7 and q8}} The picture in the isocline column represents this union of the left and right plane sets. In the &lt;math&gt;[32]R_{q7,q8}&lt;/math&gt; example it can be seen as a set of 4 Clifford parallel skew [[W:Hexagram|hexagram]]s, each having one edge in each great hexagon plane, and skewing to the left (or right) at each vertex throughout the left (or right) isoclinic rotation.{{Efn|name=clasped hands}}

== Conclusions ==
Very few if any of the observations made in this paper are original, as I hope the citations demonstrate, but some new terminology has been introduced in making them. The term '''radially equilateral''' describes a uniform polytope with its edge length equal to its long radius, because such polytopes can be constructed, with their long radii, from equilateral triangles which meet at the center, each contributing two radii and an edge. The use of the noun '''isocline''', for the circular geodesic path traced by a vertex of a 4-polytope undergoing [[#Isoclinic rotations|isoclinic rotation]], may also be new in this context. The chord-path of an isocline may be called the 4-polytope's '''Clifford polygon''', as it is the skew polygonal shape of the rotational circles traversed by the 4-polytope's vertices in its characteristic [[W:Clifford displacement|Clifford displacement]].{{Sfn|Tyrrell|Semple|1971|loc=Linear Systems of Clifford Parallels|pp=34-57}}

== Acknowledgements ==
This paper is an extract of a [[24-cell|24-cell article]] collaboratively developed by Wikipedia editors. This version contains only those sections of the Wikipedia article which I authored, or which I completely rewrote. I have removed those sections principally authored by other Wikipedia editors, and illustrations and tables which I did not create myself, except for two essential rotating animations created by Wikipedia illustrator [[Wikipedia:User:JasonHise|JasonHise]] and one by Greg Egan which I have retained with attribution. Consequently, this version is not a complete treatment of the subject; it is missing some essential topics, and it is inadequately illustrated. As a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, it is intended to gather in one place just what I have personally authored. Even so, it contains small fragments of which I am not the original author, and many editorial improvements by other Wikipedia editors. The original provenance of any sentence in this document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I am identified as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]].

Since I came to my own understanding of the 24-cell slowly, in the course of making additions to the [[Wikipedia:24-cell]] article, I am greatly indebted to the Wikipedia editors whose work on it preceded mine. Chief among these is Wikipedia editor [[W:User:Tomruen|Tomruen (Tom Ruen)]], the original author and principal illustrator of a great many of the Wikipedia articles on polytopes. The 24-cell article that I began with was already more accessible, to me, than even Coxeter's ''[[W:Regular Polytopes|Regular Polytopes]]'', or any other book treating the subject. I was inspired by the existence of Wikipedia articles on the 4-polytopes to study them more closely, and then became convinced by my own experience exploring this hypertext that the 4-polytopes could be understood much more readily, and could be documented most engagingly and comprehensively, if everything that researchers have discovered about them were incorporated into this single encyclopedic hypertext. Well-illustrated hypertext is naturally the most appropriate medium in which to describe a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning comprehension of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section of the [[Wikipedia:24-cell]] article entitled ''[[24-cell#Cell rings|Cell rings]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, also studied by Banchoff.{{Sfn|Banchoff|2013|ps=, studied the decomposition of regular 4-polytopes into honeycombs of tori tiling the [[W:Clifford torus|Clifford torus]], showed how the honeycombs correspond to [[W:Hopf fibration|Hopf fibration]]s, and made a particular study of the [[#6-cell rings|24-cell's 4 rings of 6 octahedral cells]] with illustrations.}} Finally, J.E. Mebius's definitive Wikipedia article on ''[[W:SO(4)|SO(4)]]'', the group of ''[[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]]'', informs this entire paper, which is essentially an explanation of the 24-cell's geometry as a function of its isoclinic rotations.

== Future work ==
The encyclopedia [[Wikipedia:Main_page|Wikipedia]] is not the only appropriate hypertext medium in which to explore and document the fourth dimension. Wikipedia rightly publishes only knowledge that can be sourced to previously published authorities. An encyclopedia cannot function as a research journal, in which is documented the broad, evolving edge of a field of knowledge, well before the observations made there have settled into a consensus of accepted facts. Moreover, an encyclopedia article must not become a textbook, or attempt to be the definitive whole story on a topic, or have too many footnotes! At some point in my enlargement of the [[Wikipedia:24-cell]] article, it began to transgress upon these limits, and other Wikipedia editors began to prune it back, appropriately for an encyclopedia article. I therefore sought out a home for expanded, more-than-encyclopedic versions of it and the other 4-polytope articles, where they could be enlarged by active researchers, beyond the scope of the Wikipedia encyclopedia articles.

Fortunately [[Main_page|Wikiversity]] provides just such a medium: an alternate hypertext web compatible with Wikipedia, but without the constraint of consisting of encyclopedia articles alone. A non-profit collaborative space for students and researchers, Wikiversity hosts all kinds of hypertext learning resources, such as hypertext textbooks which enlarge upon topics covered by Wikipedia, and research journals covering various fields of study which accept papers for peer review and publication. A hypertext article hosted at Wikiversity may contain links to any Wikipedia or Wikiversity article. This paper, for example, is hosted at Wikiversity, but most of its links are to Wikipedia encyclopedia articles.

Three consistent versions of the 24-cell article now exist, including this paper. The most complete version is the expanded [[24-cell]] article hosted at Wikiversity, which includes everything in the other two versions except these acknowledgments, plus additional learning resources. The original encyclopedia version, the [[Wikipedia:24-cell]] article, should be an abridged version of the expanded Wikiversity [[24-cell]] article, from which extra content inappropriate for an encyclopedia article has been removed.

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

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* {{Citation|author-last=Hise|author-first=Jason|date=2011|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a simple rotation|title-link=Wikimedia:File:24-cell.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Hise|author-first=Jason|date=2007|author-link=W:User:JasonHise|title=A 3D projection of a 24-cell performing a double rotation|title-link=Wikimedia:File:24-cell-orig.gif|journal=Wikimedia Commons}}
* {{Citation|author-last=Egan|author-first=Greg|date=2019|title=A 24-cell containing red, green, and blue 16-cells performing a double rotation|title-link=Wikimedia:File:24-cell-3CP.gif|journal=Wikimedia Commons}}
* {{Cite web|last=Egan|first=Greg|date=23 December 2021|title=Symmetries and the 24-cell|url=https://www.gregegan.net/SCIENCE/24-cell/24-cell.html|author-link=W:Greg Egan|website=gregegan.net|access-date=10 October 2022}}
{{Refend}}</text>
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  <page>
    <title>User:Atcovi/Health Psychology</title>
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      <text bytes="197" sha1="il99qcb6g2kavkntuyathc1slujo3lq" xml:space="preserve">* [[User:Atcovi/Health Psychology/Chapter 1 - What is Health?]]
* [[User:Atcovi/Health Psychology/Chapter 5 - Diverse Understandings of Stress]]

[[Category:Psychology]]

[[Category:Atcovi's Work]]</text>
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    <title>User:Atcovi/Health Psychology/Chapter 1 - What is Health?</title>
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      <text bytes="8490" sha1="hxaxqi63kaylrpxo46u2nrgjekzhz6h" xml:space="preserve">== 1.1 - What is Health? ==

=== Introduction ===

* '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]).
** How do we account for spirituality?
** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips &amp; soda (of course not!) Maintaining proper health is a daily commitment. 

=== Diversity ===

* '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans.
* Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly).
* The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. 

=== Cross-Cultural Views of Health ===

* '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). 
* The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced).  
* In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri &amp; Agnihotri, 2017, p. 31).
* Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). 
* Native Americans look at a balance between human beings and the spiritual world (nature).

== 1.2 - Defining Culture ==

=== Introduction ===

* '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto &amp; Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. 
** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. 

=== Profile of a Multicultural American ===

* America is multicultural, full of multicultural individuals.
** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. 
** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims

=== Two Key Areas of Diversity ===

* '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). 
** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). 
* '''Sex''' - One's gender.
** Many health differences, such as life expectancies, are evident between men and women.
** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues &lt;50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families).

=== Advancing Cultural Competence ===

* [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client.

== 1.3 - What is Health Psychology? ==
[[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]]

=== Introduction ===

* '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. Open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health.

=== The Evolution of Health Psychology ===

* Conceptualized as behavior medicine + medicine + array of public health sciences and services. 
* Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine).
* ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism &amp; early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. 
* '''[[Descartes]]''' comes with &quot;I think, therefore I am&quot;, further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was &quot;fine-tuned&quot; by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). 
* See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology.
* Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of originating physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942).
* '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''Psychosomatic Medicine'' for the APS.
* The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. 
* Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the &quot;frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments&quot;. '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]).

=== Health Psychology’s Biopsychosocial Approach ===

* Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior &amp; health. This describes the '''biopsychosocial approach'''. 

The field of health psychology in a nut shell:

# Stress and coping
# Health behaviors
# Issues in health care.</text>
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      <text bytes="8496" sha1="oy2r3jgdd3c4itktrxiz4k9m6invw4w" xml:space="preserve">== 1.1 - What is Health? ==

=== Introduction ===

* '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]).
** How do we account for spirituality?
** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips &amp; soda (of course not!) Maintaining proper health is a daily commitment. 

=== Diversity ===

* '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans.
* Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly).
* The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. 

=== Cross-Cultural Views of Health ===

* '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). 
* The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced).  
* In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri &amp; Agnihotri, 2017, p. 31).
* Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). 
* Native Americans look at a balance between human beings and the spiritual world (nature).

== 1.2 - Defining Culture ==

=== Introduction ===

* '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto &amp; Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. 
** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. 

=== Profile of a Multicultural American ===

* America is multicultural, full of multicultural individuals.
** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. 
** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims

=== Two Key Areas of Diversity ===

* '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). 
** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). 
* '''Sex''' - One's gender.
** Many health differences, such as life expectancies, are evident between men and women.
** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues &lt;50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families).

=== Advancing Cultural Competence ===

* [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client.

== 1.3 - What is Health Psychology? ==
[[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]]

=== Introduction ===

* '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. It is open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health.

=== The Evolution of Health Psychology ===

* Conceptualized as behavior medicine + medicine + array of public health sciences and services. 
* Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine).
* ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism &amp; early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. 
* '''[[Descartes]]''' comes with &quot;I think, therefore I am&quot;, further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was &quot;fine-tuned&quot; by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). 
* See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology.
* Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of originating physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942).
* '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''Psychosomatic Medicine'' for the APS.
* The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. 
* Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the &quot;frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments&quot;. '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]).

=== Health Psychology’s Biopsychosocial Approach ===

* Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior &amp; health. This describes the '''biopsychosocial approach'''. 

The field of health psychology in a nut shell:

# Stress and coping
# Health behaviors
# Issues in health care.</text>
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      <text bytes="8525" sha1="saeswt4h1a1s4qfraw26x0knm08qlul" xml:space="preserve">== 1.1 - What is Health? ==

=== Introduction ===

* '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]).
** How do we account for spirituality?
** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips &amp; soda (of course not!) Maintaining proper health is a daily commitment. 

=== Diversity ===

* '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans.
* Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly).
* The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. 

=== Cross-Cultural Views of Health ===

* '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). 
* The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced).  
* In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri &amp; Agnihotri, 2017, p. 31).
* Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). 
* Native Americans look at a balance between human beings and the spiritual world (nature).

== 1.2 - Defining Culture ==

=== Introduction ===

* '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto &amp; Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. 
** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. 

=== Profile of a Multicultural American ===

* America is multicultural, full of multicultural individuals.
** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. 
** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims

=== Two Key Areas of Diversity ===

* '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). 
** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). 
* '''Sex''' - One's gender.
** Many health differences, such as life expectancies, are evident between men and women.
** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues &lt;50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families).

=== Advancing Cultural Competence ===

* [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client.

== 1.3 - What is Health Psychology? ==
[[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]]

=== Introduction ===

* '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. It is open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health.

=== The Evolution of Health Psychology ===

* Conceptualized as behavior medicine + medicine + array of public health sciences and services. 
* Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine).
* ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism &amp; early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. 
* '''[[Descartes]]''' comes with &quot;I think, therefore I am&quot;, further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was &quot;fine-tuned&quot; by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). 
* See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology.
* Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of attributing physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942).
* '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''[[w:Psychosomatic_Medicine|Psychosomatic Medicine]]'' for the APS.
* The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. 
* Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the &quot;frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments&quot;. '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]).

=== Health Psychology’s Biopsychosocial Approach ===

* Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior &amp; health. This describes the '''biopsychosocial approach'''. 

The field of health psychology in a nut shell:

# Stress and coping
# Health behaviors
# Issues in health care.</text>
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    <revision>
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      <text bytes="8605" sha1="bnx88m1zola43ye21jdc6mvwgs97w73" xml:space="preserve">== 1.1 - What is Health? ==

=== Introduction ===

* '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]).
** How do we account for spirituality?
** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips &amp; soda (of course not!) Maintaining proper health is a daily commitment. 

=== Diversity ===

* '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans.
* Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly).
* The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. 

=== Cross-Cultural Views of Health ===

* '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). 
* The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced).  
* In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri &amp; Agnihotri, 2017, p. 31).
* Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). 
* Native Americans look at a balance between human beings and the spiritual world (nature).

== 1.2 - Defining Culture ==

=== Introduction ===

* '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto &amp; Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. 
** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. 

=== Profile of a Multicultural American ===

* America is multicultural, full of multicultural individuals.
** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. 
** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims

=== Two Key Areas of Diversity ===

* '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). 
** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). 
* '''Sex''' - One's gender.
** Many health differences, such as life expectancies, are evident between men and women.
** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues &lt;50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families).

=== Advancing Cultural Competence ===

* [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client.

== 1.3 - What is Health Psychology? ==
[[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]]

=== Introduction ===

* '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. It is open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health.

=== The Evolution of Health Psychology ===

* Conceptualized as behavior medicine + medicine + array of public health sciences and services. 
* Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine).
* ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism &amp; early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. 
* '''[[Descartes]]''' comes with &quot;I think, therefore I am&quot;, further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was &quot;fine-tuned&quot; by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). Philosophy played a major role in the emergence of the health psychology field. 
* See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology.
* Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of attributing physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942).
* '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''[[w:Psychosomatic_Medicine|Psychosomatic Medicine]]'' for the APS.
* The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. 
* Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the &quot;frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments&quot;. '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]).

=== Health Psychology’s Biopsychosocial Approach ===

* Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior &amp; health. This describes the '''biopsychosocial approach'''. 

The field of health psychology in a nut shell:

# Stress and coping
# Health behaviors
# Issues in health care.</text>
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      <text bytes="8675" sha1="ll40un0511a6xlfnmc9xf3btkelna25" xml:space="preserve">== 1.1 - What is Health? ==

=== Introduction ===

* '''Health''' - state of complete physical, mental, and social well-being ([[w:WHO|WHO]]).
** How do we account for spirituality?
** Could we see health as a [[w:Teeter-totter|teeter-totter]], where optimal health is on one side while poor health is on the other (defined each by our habits)? But is it really as simple as excersising frequently, yet consuming a diet of mostly chips &amp; soda (of course not!) Maintaining proper health is a daily commitment. 

=== Diversity ===

* '''Intersectionality''' - Social + political impacts = Effect on health. Death rates are higher for black Americans vs. white Americans.
* Answers about health can vary depending on '''culture''' (dynamic, but stable, set of goals, beliefs, and attitudes shared by a group of people, including sex, religion, and ethnicities) as well (between religions), financial state ($20k a year vs. $100k a year), and age (child vs. the elderly).
* The '''Association of American Medical Colleges''' tries their best to alter their recommendations towards medical educators so they can address the health disparities amongst various cultures. 

=== Cross-Cultural Views of Health ===

* '''Biomedical approach''' (commonly used in the Western world) focuses solely on the biological state of a human being (if there is no disease, then the individual is healthy!). 
* The '''Traditional Chinese Medicine (TCM)''' approach looks at health through the lens of the [[w:yin_and_yang|yin and yang philosophy]] (cold vs. hot qualities are balanced).  
* In Hindusim, the '''ayurveda''' accounts for health as “the ''three main biological units''—enzymes, tissues, and excretory functions—are in harmonious condition and when ''the mind and senses are cheerful''” (Agnihotri &amp; Agnihotri, 2017, p. 31).
* Some Mexican-Americans trust healers to cure spiritual problems, which is half of the problems that cause an illness (the other approach is essentially the biomedical approach: physical illness). 
* Native Americans look at a balance between human beings and the spiritual world (nature), and believe there are no distinctions between medicine and religion. 

== 1.2 - Defining Culture ==

=== Introduction ===

* '''Culture''' can be defined as “a unique meaning and ''information system'', ''shared by a group'' and ''transmitted across generations'', that ''allows the group to meet basic needs of survival'', by ''coordinating social behavior'' to achieve a viable existence, to ''transmit successful social behaviors'', to pursue happiness and well-being, and ''to derive meaning from life''” (Matsumoto &amp; Juang, 2017, p. 4). Can be split amongst several characteristics (Polish, a woman, rich, Jewish, for example). Culture cannot be just oversimplified as ethnicity or race. 
** Culture, in fact, contains several different features, such as ethnicity, race, religion, being tall, geographical location, athleticism, computer science major, etc. It can be several components that make you, well—''you''. 

=== Profile of a Multicultural American ===

* America is multicultural, full of multicultural individuals.
** Race: 72% White/Hispanics (Spain) or Latino (non-Spain), 13% Black, 6% Asian American, and 1% Native Americans. 
** Religion: 71% Christian, 22.8% unaffiliated, 2% Jewish, .9% Muslims

=== Two Key Areas of Diversity ===

* '''Socioeconomic status (SES)''' - An entity(ies)'s economic and social level, measured by income, occupational status, and educational level - can also influence other factors, like race and the BMI index (latter in young adults). 
** The more money you have (↑ SES), the healthier you are (through purchasing better foods, health insurance, medical services, education, etc.). 
* '''Sex''' - One's gender.
** Many health differences, such as life expectancies, are evident between men and women.
** Shaped by cultural expectations (see [[w:Drinking_culture_of_Korea#Dano]]), biological differences (estrogen and its protection against cardiovascular issues &lt;50yrs), and sociographic expectations (social roles, such as women maintaining the development of the children while men work to provide for the families).

=== Advancing Cultural Competence ===

* [[w:Purnell_Model_for_Cultural_Competence|Purnell Model for Cultural Competence]] - 12 main cultural domains a clinician should explore with a client.

== 1.3 - What is Health Psychology? ==
[[File:Bust of Hippocrates.jpg|thumb|The [[w:Hippocratic_Oath|Hippocratic Oath]] remains a big part of medicine to this day.]]

=== Introduction ===

* '''Health psychology''' is the area of psychology dedicated to the biological, psychological, and social factors behind promoting health and preventing illness. The subdivision of the APA dedicated to health psychology is known as the '''Society for Health Psychology'''. It is open to psychologists and other healthcare professionals interested in advancing the psychological aspects of mental/physical health.

=== The Evolution of Health Psychology ===

* Conceptualized as behavior medicine + medicine + array of public health sciences and services. 
* Emerged as a distinct field of study in North America in the 1960s. Came about after health professionals noticed humans were dying more from chronic diseases than other causes (such as famine).
* ''Is the mind and body connected?'' Debated for centuries. Originally seen as one and 'spirits' were the cause of illness. Taoism &amp; early Indian/Middle Eastern societies viewed them as connected. Ancient Greece challenged this notion and believed the mind and body were separate, as they valued rational thought. '''[[w:Hippocrates|Hippocrates]]''' believed in the happiness coming from the balance of four fluids. 
* '''[[Descartes]]''' comes with &quot;I think, therefore I am&quot;, further strengthening the position of the Ancient Greeks - which allowed for us to study human anatomy more deeply. '''[[w:Galen|Galen]]''' first dwelled into animal dissection to find the causes of diseases. The study of human anatomy was &quot;fine-tuned&quot; by both [[w:Andreas_Vesalius|'''Andreas Vesalius''']] (1514–1564) and the Italian artist (and the prototypical Renaissance man) [[w:Leonardo_da_Vinci|'''Leonardo da Vinci''']] (1452–1519). Descartes had to mangle with the Roman Catholic Church to allow human dissections, as he reasoned that the mind and body were separate, and this is appearant at death (when the mind and soul leaves, and the body is left). Philosophy played a major role in the emergence of the health psychology field. 
* See [[AP Psychology/Introduction]] for a brief overview of the beginning of the field of psychology.
* Psychoanalysts [[w:Franz_Alexander|'''Franz Alexander''']] and [[w:Helen_Flanders_Dunbar|'''Helen Flanders Dunbar''']] continued Sigmund Freud's work of attributing physical illness to psychological issues. They established '''[[w:Psychosomatic_medicine|psychosomatic medicine]],''' medicine dealing with the influence of minds on the health. Despite criticism for majorly holding onto Freud's beliefs (which are largely rejected/modified in today's day), the [[w:Society_for_Biopsychosocial_Science_and_Medicine|American Psychosomatic Society]] (APS) still exists to this day (founded in 1942).
* '''[[w:Behavioral_medicine|Behavioral medicine]]''' examines non-biological influences on health, such as psychological issues. '''[[w:Society_of_Behavioral_Medicine|The Society of Behavioral Medicine]] (SBM)''' was founded in 1978. The [[w:Annals_of_Behavioral_Medicine|''Annals of Behavioral Medicine'']] is the journal of the SBM, akin to ''[[w:Psychosomatic_Medicine|Psychosomatic Medicine]]'' for the APS.
* The '''[[w:International_Classification_of_Diseases|International Classification of Diseases]],''' which classifies diseases and disorders, is also a useful resource for psychologists. 
* Health psychology and medical sociology are influenced by '''[[w:Epidemiology|epidemiology]]''', a field of medicine which focuses on the &quot;frequency, distribution, and causes of different diseases with an emphasis on the role of the physical and social environments&quot;. '''Morbidity''' is the number of cases of a disease that exist during a certain period of time, and '''mortality''' is the number of deaths related to a specific cause ([textbook citation TBD]).

=== Health Psychology’s Biopsychosocial Approach ===

* Our own biological makeup, culture, society, and/or our own thoughts, behaviors, and beliefs affects our behavior &amp; health. This describes the '''biopsychosocial approach'''. 

The field of health psychology in a nut shell:

# Stress and coping
# Health behaviors
# Issues in health care.</text>
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|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
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|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
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|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
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    </revision>
    <revision>
      <id>2692067</id>
      <parentid>2692062</parentid>
      <timestamp>2024-12-15T21:04:19Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <text bytes="4934" sha1="5nnsem90faq15w72xi57qqccn6y1xez" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/style.css&quot; /&gt;

{{dh-box|code}}
&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;
|}&lt;!-- end of dh-box --&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
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|-
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|-
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|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
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|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
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|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
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|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
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    </revision>
    <revision>
      <id>2692074</id>
      <parentid>2692067</parentid>
      <timestamp>2024-12-15T21:20:26Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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      <origin>2692074</origin>
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      <text bytes="4945" sha1="m6jx5pl2jnfx7y3tyqk36b3p2605t3k" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/style.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 2|nameless 2]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
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|-
|class=&quot;size&quot;| 16
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>m6jx5pl2jnfx7y3tyqk36b3p2605t3k</sha1>
    </revision>
    <revision>
      <id>2692077</id>
      <parentid>2692074</parentid>
      <timestamp>2024-12-15T21:35:59Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692077</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4946" sha1="9njla5w48w2nwq03fsnt90ccwzie7tv" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 2|nameless 2]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
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|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
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|-
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|-
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|-
|class=&quot;size&quot;| 16
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|-
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|-
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|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>9njla5w48w2nwq03fsnt90ccwzie7tv</sha1>
    </revision>
    <revision>
      <id>2692084</id>
      <parentid>2692077</parentid>
      <timestamp>2024-12-15T22:27:59Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692084</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5106" sha1="0n0e15ujstve0mkt2j4y7quc62nxilk" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 2|nameless 2]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;


&lt;div class=&quot;intpart&quot;&gt;
Number of blocks:  &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;
Integer partition: &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 26, 40, 50, 72, 82, 96, 122, 128, 154, 168, 178, 200, 210, 224, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 27, 41, 51, 73, 83, 97, 123, 129, 155, 169, 179, 201, 211, 225, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502842589318664279370797025331501452510319371423903592348948834201632770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 24, 42, 48, 74, 80, 98, 120, 130, 152, 170, 176, 202, 208, 226, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312956423470860867487953884578135305885447208543303494136336640446038020.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 25, 43, 49, 75, 81, 99, 121, 131, 153, 171, 177, 203, 209, 227, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625912846941721734975907769156270611770894417086606988272673280892076040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 30, 44, 54, 76, 86, 100, 126, 132, 158, 172, 182, 204, 214, 228, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022740714549314234966376202652011620082554971391228738791590673613062160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 31, 45, 55, 77, 87, 101, 127, 133, 159, 173, 183, 205, 215, 229, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896045481429098628469932752405304023240165109942782457477583181347226124320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>0n0e15ujstve0mkt2j4y7quc62nxilk</sha1>
    </revision>
    <revision>
      <id>2692085</id>
      <parentid>2692084</parentid>
      <timestamp>2024-12-15T22:31:44Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692085</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5143" sha1="0paqxgc334gf281thp56gwlr097q6jm" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 2|nameless 2]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;


&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 26, 40, 50, 72, 82, 96, 122, 128, 154, 168, 178, 200, 210, 224, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 27, 41, 51, 73, 83, 97, 123, 129, 155, 169, 179, 201, 211, 225, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502842589318664279370797025331501452510319371423903592348948834201632770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 24, 42, 48, 74, 80, 98, 120, 130, 152, 170, 176, 202, 208, 226, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312956423470860867487953884578135305885447208543303494136336640446038020.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 25, 43, 49, 75, 81, 99, 121, 131, 153, 171, 177, 203, 209, 227, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625912846941721734975907769156270611770894417086606988272673280892076040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 30, 44, 54, 76, 86, 100, 126, 132, 158, 172, 182, 204, 214, 228, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022740714549314234966376202652011620082554971391228738791590673613062160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 31, 45, 55, 77, 87, 101, 127, 133, 159, 173, 183, 205, 215, 229, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896045481429098628469932752405304023240165109942782457477583181347226124320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>0paqxgc334gf281thp56gwlr097q6jm</sha1>
    </revision>
    <revision>
      <id>2692088</id>
      <parentid>2692085</parentid>
      <timestamp>2024-12-15T22:34:00Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692088</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5156" sha1="oluchlu35x4fevlkglfdtdamcouiuvq" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 2|nameless 2]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 26, 40, 50, 72, 82, 96, 122, 128, 154, 168, 178, 200, 210, 224, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 27, 41, 51, 73, 83, 97, 123, 129, 155, 169, 179, 201, 211, 225, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502842589318664279370797025331501452510319371423903592348948834201632770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 24, 42, 48, 74, 80, 98, 120, 130, 152, 170, 176, 202, 208, 226, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312956423470860867487953884578135305885447208543303494136336640446038020.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 25, 43, 49, 75, 81, 99, 121, 131, 153, 171, 177, 203, 209, 227, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625912846941721734975907769156270611770894417086606988272673280892076040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 30, 44, 54, 76, 86, 100, 126, 132, 158, 172, 182, 204, 214, 228, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022740714549314234966376202652011620082554971391228738791590673613062160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 31, 45, 55, 77, 87, 101, 127, 133, 159, 173, 183, 205, 215, 229, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896045481429098628469932752405304023240165109942782457477583181347226124320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>oluchlu35x4fevlkglfdtdamcouiuvq</sha1>
    </revision>
    <revision>
      <id>2692106</id>
      <parentid>2692088</parentid>
      <timestamp>2024-12-15T23:37:22Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692106</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5104" sha1="108u5f2nsj91ttd066oxu605vbsi3tl" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 26, 40, 50, 72, 82, 96, 122, 128, 154, 168, 178, 200, 210, 224, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 27, 41, 51, 73, 83, 97, 123, 129, 155, 169, 179, 201, 211, 225, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502842589318664279370797025331501452510319371423903592348948834201632770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 24, 42, 48, 74, 80, 98, 120, 130, 152, 170, 176, 202, 208, 226, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312956423470860867487953884578135305885447208543303494136336640446038020.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 25, 43, 49, 75, 81, 99, 121, 131, 153, 171, 177, 203, 209, 227, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625912846941721734975907769156270611770894417086606988272673280892076040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 30, 44, 54, 76, 86, 100, 126, 132, 158, 172, 182, 204, 214, 228, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022740714549314234966376202652011620082554971391228738791590673613062160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 31, 45, 55, 77, 87, 101, 127, 133, 159, 173, 183, 205, 215, 229, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896045481429098628469932752405304023240165109942782457477583181347226124320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>108u5f2nsj91ttd066oxu605vbsi3tl</sha1>
    </revision>
    <revision>
      <id>2692129</id>
      <parentid>2692106</parentid>
      <timestamp>2024-12-16T00:37:34Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692129</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5293" sha1="9evyo1onv830c76e4ypcfvjp2zufh8m" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

[[File:Set of 3-ary Boolean functions 28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|thumb|500px|similar pattern to [[Boolf prop/3-ary/patron|patron]]]]

&lt;source lang=&quot;python&quot;&gt;
x = boolf.reverse.twin(3)
val = x.quaestor(3) ^ x.consul(3)
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| nameless 1
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 26, 40, 50, 72, 82, 96, 122, 128, 154, 168, 178, 200, 210, 224, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 27, 41, 51, 73, 83, 97, 123, 129, 155, 169, 179, 201, 211, 225, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502842589318664279370797025331501452510319371423903592348948834201632770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 24, 42, 48, 74, 80, 98, 120, 130, 152, 170, 176, 202, 208, 226, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312956423470860867487953884578135305885447208543303494136336640446038020.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 25, 43, 49, 75, 81, 99, 121, 131, 153, 171, 177, 203, 209, 227, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625912846941721734975907769156270611770894417086606988272673280892076040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 30, 44, 54, 76, 86, 100, 126, 132, 158, 172, 182, 204, 214, 228, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022740714549314234966376202652011620082554971391228738791590673613062160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 31, 45, 55, 77, 87, 101, 127, 133, 159, 173, 183, 205, 215, 229, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896045481429098628469932752405304023240165109942782457477583181347226124320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 28, 46, 52, 78, 84, 102, 124, 134, 156, 174, 180, 206, 212, 230, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237007302775533773879807262153250164894167155336692855906181386247136608320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 29, 47, 53, 79, 85, 103, 125, 135, 157, 175, 181, 207, 213, 231, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474014605551067547759614524306500329788334310673385711812362772494273216640.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 18, 32, 58, 64, 90, 104, 114, 136, 146, 160, 186, 192, 218, 232, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7074290426709500931530823014194640018171510844920536908670796926665621760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 19, 33, 59, 65, 91, 105, 115, 137, 147, 161, 187, 193, 219, 233, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14148580853419001863061646028389280036343021689841073817341593853331243520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 16, 34, 56, 66, 88, 106, 112, 138, 144, 162, 184, 194, 216, 234, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1794454155477863386217099268716529896158169822403894849206313739574379520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 17, 35, 57, 67, 89, 107, 113, 139, 145, 163, 185, 195, 217, 235, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3588908310955726772434198537433059792316339644807789698412627479148759040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 22, 36, 62, 68, 94, 108, 118, 140, 150, 164, 190, 196, 222, 236, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113188646827352014904493168227114240290744173518728590538732750826649948160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 23, 37, 63, 69, 95, 109, 119, 141, 151, 165, 191, 197, 223, 237, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226377293654704029808986336454228480581488347037457181077465501653299896320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 20, 38, 60, 70, 92, 110, 116, 142, 148, 166, 188, 198, 220, 238, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28711266487645814179473588299464478338530717158462317587301019833190072320.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 21, 39, 61, 71, 93, 111, 117, 143, 149, 167, 189, 199, 221, 239, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57422532975291628358947176598928956677061434316924635174602039666380144640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 1]]</text>
      <sha1>9evyo1onv830c76e4ypcfvjp2zufh8m</sha1>
    </revision>
  </page>
  <page>
    <title>Blueprint Drawing</title>
    <ns>0</ns>
    <id>317251</id>
    <revision>
      <id>2692057</id>
      <timestamp>2024-12-15T20:46:23Z</timestamp>
      <contributor>
        <username>Ekbreckenridge</username>
        <id>2994477</id>
      </contributor>
      <comment>Redirecting Blueprint Drawing to Engineering Drawing</comment>
      <origin>2692057</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="71" sha1="f8acb3jb2ztoab71g6wywggb732m5k4" xml:space="preserve">&lt;nowiki&gt;#&lt;/nowiki&gt;REDIRECT [[Engineering Projects/Engineering Drawing]]</text>
      <sha1>f8acb3jb2ztoab71g6wywggb732m5k4</sha1>
    </revision>
    <revision>
      <id>2692061</id>
      <parentid>2692057</parentid>
      <timestamp>2024-12-15T20:52:07Z</timestamp>
      <contributor>
        <username>Ekbreckenridge</username>
        <id>2994477</id>
      </contributor>
      <minor />
      <comment>Fixing Soft Redirect</comment>
      <origin>2692061</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="57" sha1="syv88sx05ox8tc7zrxhyz0u469gex4a" xml:space="preserve">{{Softredirect|Engineering_Projects/Engineering_Drawing}}</text>
      <sha1>syv88sx05ox8tc7zrxhyz0u469gex4a</sha1>
    </revision>
  </page>
  <page>
    <title>Complex Analysis/Cauchy's Integral Theorem for Disks</title>
    <ns>0</ns>
    <id>317252</id>
    <revision>
      <id>2692058</id>
      <timestamp>2024-12-15T20:48:10Z</timestamp>
      <contributor>
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      <comment>New resource with &quot;The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the w:...&quot;</comment>
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      <text bytes="14042" sha1="c3380k5w4k4ymbinyu1z2pgin4khsfu" xml:space="preserve">The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the [[w:en:Residue theorem|Residue theorem]].

== Cauchy Integral Formula for Disks ==
=== Statement ===
Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be open, &lt;math&gt;f\colon G \to \mathbb{C}&lt;/math&gt; holomorphic, &lt;math&gt;z_0 \in G&lt;/math&gt; a point in &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;U := D_r(z_0) \subset G&lt;/math&gt; a bounded disk in &lt;math&gt;G&lt;/math&gt;. Then for all &lt;math&gt;z \in D_r(z_0)&lt;/math&gt; (i.e., for all &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;|z - z_0| &lt; r&lt;/math&gt;), the following holds: :&lt;math&gt;f(z) = \frac{1}{2\pi\mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta&lt;/math&gt; Here, &lt;math&gt;\partial U&lt;/math&gt; denotes the positively oriented curve &lt;math&gt;t \mapsto z_0 + r e^{\mathrm{i}t}&lt;/math&gt; for &lt;math&gt;t \in [0, 2\pi]&lt;/math&gt; along the boundary of the disk &lt;math&gt;U&lt;/math&gt;.
=== Proof 1 ===
For a fixed &lt;math&gt;z \in U&lt;/math&gt;, the function &lt;math&gt;g\colon U\to\mathbb{C}&lt;/math&gt; defined by &lt;math&gt;w\mapsto\tfrac{f(w)-f(z)}{w-z}&lt;/math&gt; for &lt;math&gt;w\neq z&lt;/math&gt; und &lt;math&gt;w\mapsto f'(z)&lt;/math&gt; for &lt;math&gt;w=z&lt;/math&gt;. &lt;math&gt;g&lt;/math&gt; is steadily on  &lt;math&gt;U&lt;/math&gt; and holomorphic on &lt;math&gt;U\setminus\{z\}&lt;/math&gt;.  By the [[w:en:Cauchy Integral Theorem|Cauchy Integral Theorem]], we now have: :&lt;math&gt;0 = \oint_{\partial U} g = \oint_{\partial U}\frac{f(\zeta)}{\zeta-z} \mathrm{d}\zeta - f(z)\oint_{\partial U}\frac{\mathrm{d}\zeta}{\zeta-z}&lt;/math&gt;.

=== Proof 2 ===
The function &lt;math&gt;h\colon U \to \mathbb{C}&lt;/math&gt;, &lt;math&gt;\textstyle w \mapsto \oint_{\partial U} \frac{\mathrm{d}\zeta}{\zeta-w}&lt;/math&gt; is holomorphic with the derivative &lt;math&gt;\textstyle h'(w) = \oint_{\partial U} \frac{\mathrm{d}\zeta}{\left(\zeta-w\right)^2}&lt;/math&gt;, which vanishes since the integrand has an antiderivative (namely &lt;math&gt;\zeta \mapsto -\frac{1}{\zeta-w}&lt;/math&gt;). Therefore, &lt;math&gt;h&lt;/math&gt; is constant, and since &lt;math&gt;h(a) = 2\pi i&lt;/math&gt;, we have &lt;math&gt;h(z) = 2\pi i&lt;/math&gt;.

== Consequences of the Cauchy Integral Theorem ==
The Cauchy Integral Theorem (CIS) leads to the following corollaries:
=== Representation of the Function at the Center of the Disk ===
For every holomorphic function, the function value at the center of a circle is the average of the function values on the circle's boundary. Use &lt;math&gt;\zeta(t) = z_o + r e^{\mathrm{i}t},\ \mathrm{d}\zeta = \mathrm{i} r e^{\mathrm{i}t} \mathrm{d}t&lt;/math&gt;.
Test: :&lt;math&gt; \begin{align} f|{U}(z_o) &amp;= \frac{1}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{\zeta - z_o} \mathrm{d}\zeta = \frac{1}{2\pi \mathrm{i}} \int_{0}^{2\pi} \frac{f(a + r e^{\mathrm{i}t})}{r e^{\mathrm{i}t}} \mathrm{i} r e^{\mathrm{i}t} , \mathrm{d}t \ &amp;= \frac{1}{2\pi} \int_{0}^{2\pi} f(z_o + r e^{\mathrm{i}t}) , \mathrm{d}t \end{align}&lt;/math&gt;

=== Derivatives - Cauchy Integral Formula - CIF ===
Every holomorphic function is infinitely complex differentiable, and each of these derivatives is also holomorphic. Expressed using the integral formula, this means for &lt;math&gt;|z - z_o| &lt; r&lt;/math&gt; and &lt;math&gt;n \in \mathbb{N}{0}&lt;/math&gt;: :&lt;math&gt;f^{(n)}(z) = \frac{n!}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{(\zeta - z)^{n+1}} \mathrm{d}\zeta.&lt;/math&gt;
=== Local Developability in Power Series ===
Every holomorphic function can be locally expanded into a [[w:en:Power Series|power series]] for &lt;math&gt;|z - a| &lt; r&lt;/math&gt;.
:&lt;math&gt;f(z) = \sum\limits_{n=0}^\infty \left( \frac{1}{2\pi \mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{(\zeta - a)^{n+1}} \mathrm{d}\zeta \right) (z - a)^n = \sum\limits_{n=0}^\infty a_n (z - a)^n.&lt;/math&gt;
Using the integral formula for &lt;math&gt;f^{(n)}&lt;/math&gt;, it immediately follows that the coefficients &lt;math&gt;a_n&lt;/math&gt; are exactly the [[w:en:Taylor series|Taylor coefficients]].

=== Estimation of the Taylor Series Coefficients ===
For the coefficients, the following estimate holds when &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - a| &lt; r \ \Leftrightarrow z \in U_r(a)&lt;/math&gt;: :&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
The [[w:en:Liouville's Theorem|Liouville Theorem]] (every [[w:en:Entire Function|holomorphic function bounded on the entire complex plane]] is constant) can be easily proven using the integral formula. This can then be used to easily prove the [[w:en:Fundamental Theorem of Algebra|Fundamental Theorem of Algebra]] (every polynomial in &lt;math&gt;\mathbb{C}&lt;/math&gt; factors into linear factors).
Here's the translation with the specified conditions:

=== Proof 1 ===
The Cauchy integral formula is differentiated partially, allowing differentiation and integration to be swapped:
:&lt;math&gt;\begin{align} f^{(n)}|_{U}(z) &amp; =\frac{\partial^{n}f}{\partial z^{n}}|_{U}(z)=\frac{1}{2\pi\mathrm{i}}\frac{\partial^{n}}{\partial z^{n}}\oint_{\partial U}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta\\  &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial U}f(\zeta)\underbrace{\frac{\partial^{n}}{\partial z^{n}}\frac{1}{\zeta-z}}_{n!/(\zeta-z)^{1+n}}\mathrm{d}\zeta=\frac{n!}{2\pi\mathrm{i}}\oint_{\partial U}\frac{f(\zeta)}{(\zeta-z)^{1+n}}\mathrm{d}\zeta\end{align} &lt;/math&gt;
=== Proof 2a: Cauchy Kernel ===
Developing &lt;math&gt;\frac{1}{\zeta - z}&lt;/math&gt; in the Cauchy integral formula using the geometric series gives (Cauchy kernel):
:&lt;math&gt; \frac{1}{1 - \frac{z - z_o}{\zeta - z_o}} = \sum_{n=0}^{\infty} \left( \frac{z - z_o}{\zeta - z_o} \right)^{n} &lt;/math&gt;

=== Proof 2: Cauchy Kernel - Taylor Series ===
:&lt;math&gt;\begin{align} f|_{U}(z) &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ &amp; {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &amp;\overset{|\frac{z-z_o}{\zeta-z_o}|&lt;1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ &amp; =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}&lt;/math&gt;

=== Proof 2b: Cauchy Kernel ===
Since the geometric series converges uniformly for &lt;math&gt;|z - z_o| &lt; |\zeta - z_o| = r&lt;/math&gt;, one can integrate term by term, i.e., swap the sum and the integral. The development coefficients are:
:&lt;math&gt;\begin{align} a_{n} &amp; =\frac{1}{n!}f^{(n)}|_{U}(z_o)=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_{r}(a)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\\ &amp; =\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f(z_o+re^{\mathrm{i}t})}{(re^{\mathrm{i}t})^{n+1}}\mathrm{i}re^{\mathrm{i}t}\,\mathrm{d}t=\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\end{align}&lt;/math&gt;

=== Proof 3: Estimation of the Coefficients ===
For the coefficients &lt;math&gt;a_n \in \mathbb{C}&lt;/math&gt;, the following estimate holds. There exists a &lt;math&gt;M &gt; 0&lt;/math&gt; such that &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - z_o| = r&lt;/math&gt;. Then, for &lt;math&gt;n \in \mathbb{N}0&lt;/math&gt;, we have:
:&lt;math&gt;\begin{align} |a_{n}|&amp;=\left|\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\right|\\ &amp;\leq\frac{1}{2\pi r^n}\int_0^{2\pi}\underbrace{|f(z_o+re^{\mathrm{i} t})|}_{\leq M}\,\mathrm{d}t\leq \frac{M}{r^{n}}\end{align}&lt;/math&gt;

=== Proof 4: Liouville's Theorem ===
If &lt;math&gt;f&lt;/math&gt; is holomorphic on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; and bounded, i.e., &lt;math&gt;|f(z)| = |\sum_{n=0}^{\infty} a_n z^n| \leq M&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;, then, as before, for all &lt;math&gt;r &gt; 0&lt;/math&gt;, we have:
:&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
Since &lt;math&gt;r&lt;/math&gt; was arbitrary, it follows that &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb{N}&lt;/math&gt;. Therefore, from the boundedness of &lt;math&gt;f&lt;/math&gt;, we conclude:
: &lt;math&gt;f(z) = a_0&lt;/math&gt;
Thus, every bounded holomorphic function on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; is constant (Liouville's theorem).

=== Example ===
Using the integral formula, integrals can also be computed:
:&lt;math&gt; \oint_{\partial U_2(0)} \frac{e^{2\zeta}}{(\zeta + 1)^4} \mathrm{d}\zeta = \frac{2\pi \mathrm{i}}{3!} \frac{\mathrm{d}^3}{\mathrm{d}z^3} e^{2z} |_{z = -1} = \frac{8 \pi \mathrm{i}}{3 e^2} &lt;/math&gt;

== Cauchy Integral Formula for Cycles ==
A generalization of the integral formula for circular contours is the version for cycles:

Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be a domain, &lt;math&gt;f \colon G \to \mathbb{C}&lt;/math&gt; holomorphic, and &lt;math&gt;\Gamma&lt;/math&gt; a [[w:en:zero homologous|zero homologous]] [[w:en:cycle|cycle]] in &lt;math&gt;D&lt;/math&gt;. Then, for all &lt;math&gt;z \in D&lt;/math&gt; not on &lt;math&gt;\Gamma&lt;/math&gt;, the following integral formula holds:
:&lt;math&gt; n(\Gamma, z) \cdot f(z) = \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta &lt;/math&gt;

Here, &lt;math&gt;n(\Gamma, z)&lt;/math&gt; denotes the [[w:en:winding number|winding number]] or [[w:en:revolution|revolution]] of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.

== Cauchy Integral Formula for Polycycles ==
The Cauchy integral formula has been generalized to the multidimensional complex space &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;U_1, \ldots, U_n&lt;/math&gt; be disk domains in &lt;math&gt;\mathbb{C}&lt;/math&gt;, then &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt; is a [[w:en:Polycylinder|Polycylinder]] in &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; be a holomorphic function and &lt;math&gt;\xi \in U&lt;/math&gt;. The Cauchy integral formula is given by
:&lt;math&gt; f(z_1, \ldots, z_n) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U_n} \cdots \oint_{\partial U_1} \frac{f(\xi_1, \ldots, \xi_n)}{(\xi_1 - z_1) \cdots (\xi_n - z_n)} \mathrm{d} \xi_1 \cdots \mathrm{d} \xi_n &lt;/math&gt;

=== Restrictions in Multidimensional Space ===
Since the Cauchy integral theorem does not hold in higher-dimensional space, this formula cannot be derived analogously to the one-dimensional case. Therefore, this integral formula is derived using [[w:en:Induction (Mathematik)|induction]] from the Cauchy integral formula for disk domains. Using the [[w:en:Multiindex|multi-index]] notation, the formula can be simplified to:
:&lt;math&gt; f(z) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U} \frac{f(\xi)}{(\xi - z)} , \mathrm{d} \xi &lt;/math&gt;
with &lt;math&gt;\partial U = \partial U_1 \times \cdots \times \partial U_n&lt;/math&gt;.

=== Polycycles ===
Polycycles are defined using a vector of radii, where &lt;math&gt; M := \max_{\xi \in U} |f(\xi)| &lt;/math&gt; and &lt;math&gt; r = (r_1, \ldots, r_n) &lt;/math&gt; is the radius of the polycycle &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt;.&lt;ref&gt;
for the derivatives of the holomorphic Function &lt;math&gt;f&lt;/math&gt; as well as Cauchy's inequality 
:&lt;math&gt;\left|D^k f(z)\right |\le \frac{M \cdot k!}{r^k},&lt;/math&gt;

== See also ==

*[[cycle]]

*[[Cauchy's Integral Theorem for Cycles]]

*[[null-homologous|zero homologous]]

== References == 
 &lt;references /&gt;

== Literature ==
*Kurt Endl, [[w:de:Wolfgang Luh|Wolfgang Luh]]: ''Analysis.'' Volume 3: ''Function Theory, Differential Equations.'' 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9, p. 153, Theorem 4.9.1.
*Wolfgang Fischer, [[w:de:Ingo Lieb|Ingo Lieb]]: ''Function Theory.'' 7th improved edition. Vieweg, Braunschweig, 1994, ISBN 3-528-67247-1, p. 60, Chapter 3, Theorem 2.2 (''Vieweg-Studium. Advanced Mathematics Course'' 47).
[[Category: Function Theory]] [[Category: Theorem (Mathematics)|Cauchy's Integral Formula]]
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      <text bytes="14042" sha1="98psfrurj3zgulv0annd6b2gjxym4vo" xml:space="preserve">The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the [[w:en:Residue theorem|Residue theorem]].

== Cauchy Integral Formula for Disks ==
=== Statement ===
Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be open, &lt;math&gt;f\colon G \to \mathbb{C}&lt;/math&gt; holomorphic, &lt;math&gt;z_0 \in G&lt;/math&gt; a point in &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;U := D_r(z_0) \subset G&lt;/math&gt; a bounded disk in &lt;math&gt;G&lt;/math&gt;. Then for all &lt;math&gt;z \in D_r(z_0)&lt;/math&gt; (i.e., for all &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;|z - z_0| &lt; r&lt;/math&gt;), the following holds: :&lt;math&gt;f(z) = \frac{1}{2\pi\mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta&lt;/math&gt; Here, &lt;math&gt;\partial U&lt;/math&gt; denotes the positively oriented curve &lt;math&gt;t \mapsto z_0 + r e^{\mathrm{i}t}&lt;/math&gt; for &lt;math&gt;t \in [0, 2\pi]&lt;/math&gt; along the boundary of the disk &lt;math&gt;U&lt;/math&gt;.
=== Proof 1 ===
For a fixed &lt;math&gt;z \in U&lt;/math&gt;, the function &lt;math&gt;g\colon U\to\mathbb{C}&lt;/math&gt; defined by &lt;math&gt;w\mapsto\tfrac{f(w)-f(z)}{w-z}&lt;/math&gt; for &lt;math&gt;w\neq z&lt;/math&gt; und &lt;math&gt;w\mapsto f'(z)&lt;/math&gt; for &lt;math&gt;w=z&lt;/math&gt;. &lt;math&gt;g&lt;/math&gt; is steadily on  &lt;math&gt;U&lt;/math&gt; and holomorphic on &lt;math&gt;U\setminus\{z\}&lt;/math&gt;.  By the [[w:en:Cauchy Integral Theorem|Cauchy Integral Theorem]], we now have: :&lt;math&gt;0 = \oint_{\partial U} g = \oint_{\partial U}\frac{f(\zeta)}{\zeta-z} \mathrm{d}\zeta - f(z)\oint_{\partial U}\frac{\mathrm{d}\zeta}{\zeta-z}&lt;/math&gt;.

=== Proof 2 ===
The function &lt;math&gt;h\colon U \to \mathbb{C}&lt;/math&gt;, &lt;math&gt;\textstyle w \mapsto \oint_{\partial U} \frac{\mathrm{d}\zeta}{\zeta-w}&lt;/math&gt; is holomorphic with the derivative &lt;math&gt;\textstyle h'(w) = \oint_{\partial U} \frac{\mathrm{d}\zeta}{\left(\zeta-w\right)^2}&lt;/math&gt;, which vanishes since the integrand has an antiderivative (namely &lt;math&gt;\zeta \mapsto -\frac{1}{\zeta-w}&lt;/math&gt;). Therefore, &lt;math&gt;h&lt;/math&gt; is constant, and since &lt;math&gt;h(a) = 2\pi i&lt;/math&gt;, we have &lt;math&gt;h(z) = 2\pi i&lt;/math&gt;.

== Consequences of the Cauchy Integral Theorem ==
The Cauchy Integral Theorem (CIS) leads to the following corollaries:
=== Representation of the Function at the Center of the Disk ===
For every holomorphic function, the function value at the center of a circle is the average of the function values on the circle's boundary. Use &lt;math&gt;\zeta(t) = z_o + r e^{\mathrm{i}t},\ \mathrm{d}\zeta = \mathrm{i} r e^{\mathrm{i}t} \mathrm{d}t&lt;/math&gt;.
Test: :&lt;math&gt; \begin{align} f|{U}(z_o) &amp;= \frac{1}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{\zeta - z_o} \mathrm{d}\zeta = \frac{1}{2\pi \mathrm{i}} \int_{0}^{2\pi} \frac{f(a + r e^{\mathrm{i}t})}{r e^{\mathrm{i}t}} \mathrm{i} r e^{\mathrm{i}t} , \mathrm{d}t \ &amp;= \frac{1}{2\pi} \int_{0}^{2\pi} f(z_o + r e^{\mathrm{i}t}) , \mathrm{d}t \end{align}&lt;/math&gt;

=== Derivatives - Cauchy Integral Formula - CIF ===
Every holomorphic function is infinitely complex differentiable, and each of these derivatives is also holomorphic. Expressed using the integral formula, this means for &lt;math&gt;|z - z_o| &lt; r&lt;/math&gt; and &lt;math&gt;n \in \mathbb{N}{0}&lt;/math&gt;: :&lt;math&gt;f^{(n)}(z) = \frac{n!}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{(\zeta - z)^{n+1}} \mathrm{d}\zeta.&lt;/math&gt;
=== Local Developability in Power Series ===
Every holomorphic function can be locally expanded into a [[w:en:Power Series|power series]] for &lt;math&gt;|z - a| &lt; r&lt;/math&gt;.
:&lt;math&gt;f(z) = \sum\limits_{n=0}^\infty \left( \frac{1}{2\pi \mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{(\zeta - a)^{n+1}} \mathrm{d}\zeta \right) (z - a)^n = \sum\limits_{n=0}^\infty a_n (z - a)^n.&lt;/math&gt;
Using the integral formula for &lt;math&gt;f^{(n)}&lt;/math&gt;, it immediately follows that the coefficients &lt;math&gt;a_n&lt;/math&gt; are exactly the [[w:en:Taylor series|Taylor coefficients]].

=== Estimation of the Taylor Series Coefficients ===
For the coefficients, the following estimate holds when &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - a| &lt; r \ \Leftrightarrow z \in U_r(a)&lt;/math&gt;: :&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
The [[w:en:Liouville's Theorem|Liouville Theorem]] (every [[w:en:Entire Function|holomorphic function bounded on the entire complex plane]] is constant) can be easily proven using the integral formula. This can then be used to easily prove the [[w:en:Fundamental Theorem of Algebra|Fundamental Theorem of Algebra]] (every polynomial in &lt;math&gt;\mathbb{C}&lt;/math&gt; factors into linear factors).
Here's the translation with the specified conditions:

=== Proof 1 ===
The Cauchy integral formula is differentiated partially, allowing differentiation and integration to be swapped:
:&lt;math&gt;\begin{align} f^{(n)}|_{U}(z) &amp; =\frac{\partial^{n}f}{\partial z^{n}}|_{U}(z)=\frac{1}{2\pi\mathrm{i}}\frac{\partial^{n}}{\partial z^{n}}\oint_{\partial U}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta\\  &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial U}f(\zeta)\underbrace{\frac{\partial^{n}}{\partial z^{n}}\frac{1}{\zeta-z}}_{n!/(\zeta-z)^{1+n}}\mathrm{d}\zeta=\frac{n!}{2\pi\mathrm{i}}\oint_{\partial U}\frac{f(\zeta)}{(\zeta-z)^{1+n}}\mathrm{d}\zeta\end{align} &lt;/math&gt;
=== Proof 2a: Cauchy Kernel ===
Developing &lt;math&gt;\frac{1}{\zeta - z}&lt;/math&gt; in the Cauchy integral formula using the geometric series gives (Cauchy kernel):
:&lt;math&gt; \frac{1}{1 - \frac{z - z_o}{\zeta - z_o}} = \sum_{n=0}^{\infty} \left( \frac{z - z_o}{\zeta - z_o} \right)^{n} &lt;/math&gt;

=== Proof 2: Cauchy Kernel - Taylor Series ===
:&lt;math&gt;\begin{align} f|_{U}(z) &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ &amp; {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &amp;\overset{|\frac{z-z_o}{\zeta-z_o}|&lt;1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ &amp; =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}&lt;/math&gt;

=== Proof 2b: Cauchy Kernel ===
Since the geometric series converges uniformly for &lt;math&gt;|z - z_o| &lt; |\zeta - z_o| = r&lt;/math&gt;, one can integrate term by term, i.e., swap the sum and the integral. The development coefficients are:
:&lt;math&gt;\begin{align} a_{n} &amp; =\frac{1}{n!}f^{(n)}|_{U}(z_o)=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_{r}(a)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\\ &amp; =\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f(z_o+re^{\mathrm{i}t})}{(re^{\mathrm{i}t})^{n+1}}\mathrm{i}re^{\mathrm{i}t}\,\mathrm{d}t=\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\end{align}&lt;/math&gt;

=== Proof 3: Estimation of the Coefficients ===
For the coefficients &lt;math&gt;a_n \in \mathbb{C}&lt;/math&gt;, the following estimate holds. There exists a &lt;math&gt;M &gt; 0&lt;/math&gt; such that &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - z_o| = r&lt;/math&gt;. Then, for &lt;math&gt;n \in \mathbb{N}0&lt;/math&gt;, we have:
:&lt;math&gt;\begin{align} |a_{n}|&amp;=\left|\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\right|\\ &amp;\leq\frac{1}{2\pi r^n}\int_0^{2\pi}\underbrace{|f(z_o+re^{\mathrm{i} t})|}_{\leq M}\,\mathrm{d}t\leq \frac{M}{r^{n}}\end{align}&lt;/math&gt;

=== Proof 4: Liouville's Theorem ===
If &lt;math&gt;f&lt;/math&gt; is holomorphic on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; and bounded, i.e., &lt;math&gt;|f(z)| = |\sum_{n=0}^{\infty} a_n z^n| \leq M&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;, then, as before, for all &lt;math&gt;r &gt; 0&lt;/math&gt;, we have:
:&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
Since &lt;math&gt;r&lt;/math&gt; was arbitrary, it follows that &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb{N}&lt;/math&gt;. Therefore, from the boundedness of &lt;math&gt;f&lt;/math&gt;, we conclude:
: &lt;math&gt;f(z) = a_0&lt;/math&gt;
Thus, every bounded holomorphic function on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; is constant (Liouville's theorem).

=== Example ===
Using the integral formula, integrals can also be computed:
:&lt;math&gt; \oint_{\partial U_2(0)} \frac{e^{2\zeta}}{(\zeta + 1)^4} \mathrm{d}\zeta = \frac{2\pi \mathrm{i}}{3!} \frac{\mathrm{d}^3}{\mathrm{d}z^3} e^{2z} |_{z = -1} = \frac{8 \pi \mathrm{i}}{3 e^2} &lt;/math&gt;

== Cauchy Integral Formula for Cycles ==
A generalization of the integral formula for circular contours is the version for cycles:

Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be a domain, &lt;math&gt;f \colon G \to \mathbb{C}&lt;/math&gt; holomorphic, and &lt;math&gt;\Gamma&lt;/math&gt; a [[w:en:zero homologous|zero homologous]] [[w:en:cycle|cycle]] in &lt;math&gt;D&lt;/math&gt;. Then, for all &lt;math&gt;z \in D&lt;/math&gt; not on &lt;math&gt;\Gamma&lt;/math&gt;, the following integral formula holds:
:&lt;math&gt; n(\Gamma, z) \cdot f(z) = \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta &lt;/math&gt;

Here, &lt;math&gt;n(\Gamma, z)&lt;/math&gt; denotes the [[w:en:winding number|winding number]] or [[w:en:revolution|revolution]] of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.

== Cauchy Integral Formula for Polycycles ==
The Cauchy integral formula has been generalized to the multidimensional complex space &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;U_1, \ldots, U_n&lt;/math&gt; be disk domains in &lt;math&gt;\mathbb{C}&lt;/math&gt;, then &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt; is a [[w:en:Polycylinder|Polycylinder]] in &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; be a holomorphic function and &lt;math&gt;\xi \in U&lt;/math&gt;. The Cauchy integral formula is given by
:&lt;math&gt; f(z_1, \ldots, z_n) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U_n} \cdots \oint_{\partial U_1} \frac{f(\xi_1, \ldots, \xi_n)}{(\xi_1 - z_1) \cdots (\xi_n - z_n)} \mathrm{d} \xi_1 \cdots \mathrm{d} \xi_n &lt;/math&gt;

=== Restrictions in Multidimensional Space ===
Since the Cauchy integral theorem does not hold in higher-dimensional space, this formula cannot be derived analogously to the one-dimensional case. Therefore, this integral formula is derived using [[w:en:Induction (Mathematik)|induction]] from the Cauchy integral formula for disk domains. Using the [[w:en:Multiindex|multi-index]] notation, the formula can be simplified to:
:&lt;math&gt; f(z) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U} \frac{f(\xi)}{(\xi - z)} , \mathrm{d} \xi &lt;/math&gt;
with &lt;math&gt;\partial U = \partial U_1 \times \cdots \times \partial U_n&lt;/math&gt;.

=== Polycycles ===
Polycycles are defined using a vector of radii, where &lt;math&gt; M := \max_{\xi \in U} |f(\xi)| &lt;/math&gt; and &lt;math&gt; r = (r_1, \ldots, r_n) &lt;/math&gt; is the radius of the polycycle &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt;.&lt;ref&gt;
for the derivatives of the holomorphic Function &lt;math&gt;f&lt;/math&gt; as well as Cauchy's inequality 
:&lt;math&gt;\left|D^k f(z)\right |\le \frac{M \cdot k!}{r^k},&lt;/math&gt;

== See also ==

*[[cycle]]

*[[Cauchy's Integral Theorem for Cycles]]

*[[null-homologous|zero homologous]]

== References == 
 &lt;references /&gt;

== Literature ==
*Kurt Endl, [[w:de:Wolfgang Luh|Wolfgang Luh]]: ''Analysis.'' Volume 3: ''Function Theory, Differential Equations.'' 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9, p. 153, Theorem 4.9.1.
*Wolfgang Fischer, [[w:de:Ingo Lieb|Ingo Lieb]]: ''Function Theory.'' 7th improved edition. Vieweg, Braunschweig, 1994, ISBN 3-528-67247-1, p. 60, Chapter 3, Theorem 2.2 (''Vieweg-Studium. Advanced Mathematics Course'' 47).
[[Category: Function Theory]] [[Category: Theorem (Mathematics)|Cauchy's Integral Formula]]
== Page Information ==
The following information explains how this page was created and why the source from Wikipedia was modified using the [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity converter] to [https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Cauchy_Integralsatz_f%C3%BCr_Kreisscheiben&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes Wiki2Reveal] for use as an online presentation.

=== Wikipedia2Wikiversity ===
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* Date: 21.12.2018
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|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
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|-
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|-
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[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|-
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|-
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|-
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[[Category:Boolf prop/3-ary| ]]</text>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|-
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|-
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|-
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|}

[[Category:Boolf prop/3-ary| ]]</text>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
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|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
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|}

[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|-
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|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
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[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|-
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|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
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|-
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|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
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|-
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|-
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|-
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|-
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|-
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|-
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|-
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|}

[[Category:Boolf prop/3-ary| ]]</text>
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|-
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! properties
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|-
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|-
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|-
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|-
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|-
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|-
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|}

[[Category:Boolf prop/3-ary| ]]</text>
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|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    <revision>
      <id>2692130</id>
      <parentid>2692127</parentid>
      <timestamp>2024-12-16T00:40:05Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    <revision>
      <id>2692133</id>
      <parentid>2692130</parentid>
      <timestamp>2024-12-16T00:44:32Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    <revision>
      <id>2692149</id>
      <parentid>2692133</parentid>
      <timestamp>2024-12-16T10:32:47Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/guild|guild]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    <revision>
      <id>2692152</id>
      <parentid>2692149</parentid>
      <timestamp>2024-12-16T10:45:46Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great guild|great guild]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/guild|guild]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692154</id>
      <parentid>2692152</parentid>
      <timestamp>2024-12-16T10:48:36Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
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{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
!class=&quot;unsortable&quot;| integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/greater guild|greater guild]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great guild|great guild]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/guild|guild]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692165</id>
      <parentid>2692154</parentid>
      <timestamp>2024-12-16T11:35:47Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692165</origin>
      <model>wikitext</model>
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      <text bytes="3712" sha1="hnmssb1wl67ndpctgxf3dylzs3f067s" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/greater guild|greater guild]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great guild|great guild]]
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great principality|great principality]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/guild|guild]]
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/principality|principality]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
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    </revision>
    <revision>
      <id>2692167</id>
      <parentid>2692165</parentid>
      <timestamp>2024-12-16T11:40:02Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692167</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4548" sha1="mgvhrzp0uz0b35qf75c5jinshfuyr1v" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/greater guild|greater guild]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great patron|great patron]], patron tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great quaestor|great quaestor]], quaestor tiling and slatting
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great prefect|great prefect]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great praetor|great praetor]]
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quadrant|quadrant]]
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great guild|great guild]]
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great principality|great principality]]
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/great dominion|great dominion]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 1|nameless 1]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/prefect|prefect]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/praetor|praetor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/quaestor|quaestor]]
|-
|class=&quot;number-of-blocks&quot;| 16
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/patron|patron]], patron index
|-
|class=&quot;number-of-blocks&quot;| 20
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/guild|guild]]
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/principality|principality]]
|-
|class=&quot;number-of-blocks&quot;| 44
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/dominion|dominion]]
|-
|class=&quot;number-of-blocks&quot;| 64
|class=&quot;intpart&quot;| &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
|class=&quot;props&quot;| [[Boolf prop/3-ary/nameless 2|nameless 2]]
|}

[[Category:Boolf prop/3-ary| ]]</text>
      <sha1>mgvhrzp0uz0b35qf75c5jinshfuyr1v</sha1>
    </revision>
    <revision>
      <id>2692171</id>
      <parentid>2692167</parentid>
      <timestamp>2024-12-16T11:53:48Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692171</origin>
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      <text bytes="6479" sha1="4101k0qdhnu4wof1l94d2i1cku677vc" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/props.css&quot; /&gt;
{| class=&quot;wikitable sortable boolf-props&quot; style=&quot;text-align: center;&quot;
|-
! &lt;abbr title=&quot;number of blocks&quot;&gt;#&lt;/abbr&gt;
! integer partition
! properties
|-
|class=&quot;number-of-blocks&quot;| 3
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[80, 2, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/greater guild|greater guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great patron|great patron]]&lt;/span&gt;&lt;span class=&quot;prop&quot;&gt;patron tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great quaestor|great quaestor]]&lt;/span&gt;&lt;span class=&quot;prop&quot;&gt;quaestor tiling and slatting&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great prefect|great prefect]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 4
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great praetor|great praetor]]&lt;/span&gt;
|-
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|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[64, 4]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quadrant|quadrant]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 5
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[40, 4, 96, 1]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great guild|great guild]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great principality|great principality]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 11
|class=&quot;intpart&quot;| &lt;span class=&quot;sortkey&quot;&gt;[4, 2, 12, 2, 16, 2, 24, 2, 48, 3]&lt;/span&gt;&lt;span class=&quot;formula&quot;&gt;&lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;48&lt;/span&gt;&lt;/span&gt;
|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/great dominion|great dominion]]&lt;/span&gt;
|-
|class=&quot;number-of-blocks&quot;| 16
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/nameless 1|nameless 1]]&lt;/span&gt;
|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/prefect|prefect]]&lt;/span&gt;
|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/quaestor|quaestor]]&lt;/span&gt;
|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop&quot;&gt;patron index&lt;/span&gt;
|-
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|-
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|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
|}

[[Category:Boolf prop/3-ary| ]]</text>
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  <page>
    <title>User:Atcovi/Health Psychology/Chapter 5 - Diverse Understandings of Stress</title>
    <ns>2</ns>
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      <id>2692070</id>
      <timestamp>2024-12-15T21:11:20Z</timestamp>
      <contributor>
        <username>Atcovi</username>
        <id>276019</id>
      </contributor>
      <comment>Create</comment>
      <origin>2692070</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="18" sha1="ra7zlxlb04d1bim5jvsevzinozkrnby" xml:space="preserve">== Introduction ==</text>
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    <title>Boolf prop/3-ary/nameless 2</title>
    <ns>0</ns>
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    <revision>
      <id>2692071</id>
      <timestamp>2024-12-15T21:11:48Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/style.css&quot; /&gt;  compare [[Boolf prop/3-ary/nameless 1|nameless 1]]  &lt;source lang=&quot;python&quot;&gt; val = (boolf ^ boolf.mentor(3).twin(3)).zhe &lt;/source&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| nameless 2 !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 4 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 40, 72, 96]&lt;/span&gt;File:Set_of_3-ary_Boolean_functions_79228167236630821562...&quot;</comment>
      <origin>2692071</origin>
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      <text bytes="13747" sha1="5sumc0uhveft3x7b1qj045cscl94zz9" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/style.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 1|nameless 1]]

&lt;source lang=&quot;python&quot;&gt;
val = (boolf ^ boolf.mentor(3).twin(3)).zhe
&lt;/source&gt;

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[[Category:Boolf prop/3-ary|nameless 2]]</text>
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    <revision>
      <id>2692078</id>
      <parentid>2692071</parentid>
      <timestamp>2024-12-15T21:36:08Z</timestamp>
      <contributor>
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        <id>137431</id>
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compare [[Boolf prop/3-ary/nameless 1|nameless 1]]

&lt;source lang=&quot;python&quot;&gt;
val = (boolf ^ boolf.mentor(3).twin(3)).zhe
&lt;/source&gt;

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[[Category:Boolf prop/3-ary|nameless 2]]</text>
      <sha1>jnhwohm62gwteu6yyihw99b28jaj634</sha1>
    </revision>
    <revision>
      <id>2692090</id>
      <parentid>2692078</parentid>
      <timestamp>2024-12-15T22:38:49Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692090</origin>
      <model>wikitext</model>
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      <text bytes="13957" sha1="3pc3slw52o3kf025jh4w99o5r5o9kjw" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

compare [[Boolf prop/3-ary/nameless 1|nameless 1]]

&lt;source lang=&quot;python&quot;&gt;
val = (boolf ^ boolf.mentor(3).twin(3)).zhe
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
&lt;/div&gt;

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[[Category:Boolf prop/3-ary|nameless 2]]</text>
      <sha1>3pc3slw52o3kf025jh4w99o5r5o9kjw</sha1>
    </revision>
    <revision>
      <id>2692107</id>
      <parentid>2692090</parentid>
      <timestamp>2024-12-15T23:40:38Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692107</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="14133" sha1="1zlgxt8p9q4stlyrccj8i01n6w84p4t" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

[[File:Set of 3-ary Boolean functions 87183322370842425795587566660030343945541236792690890543826997033971843334915.svg|thumb|500px|The values 0, 1, 8, 9... form the block of [[Boolf prop/3-ary/great patron|great patron]] 0.]]

&lt;source lang=&quot;python&quot;&gt;
val = (boolf ^ boolf.mentor(3).twin(3)).zhe
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;64&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt;
&lt;/div&gt;

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|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[140, 164, 196, 236]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_110427941548749454226745664716244870891588489121053265490066829647806464.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 182
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[141, 165, 197, 237]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_220855883097498908453491329432489741783176978242106530980133659295612928.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 31
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[142, 166, 198, 238]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_441711766194997816906982658864979483566353956484213061960267318591225856.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 119
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[143, 167, 199, 239]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_883423532389995633813965317729958967132707912968426123920534637182451712.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 224
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[144, 184, 216, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766847170090676022660412852537468678429999273317197014843191772047212544.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 136
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[145, 185, 217, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3533694340181352045320825705074937356859998546634394029686383544094425088.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 33
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[146, 186, 218, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7067388680362704090641651410149874713719997093268788059372767088188850176.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 73
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[147, 187, 219, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14134777360725408181283302820299749427439994186537576118745534176377700352.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 65
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[148, 188, 220, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28269554721450816362566605640599498854879988373075152237491068352755400704.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 41
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[149, 189, 221, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56539109442901632725133211281198997709759976746150304474982136705510801408.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[150, 190, 222, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113078218885803265450266422562397995419519953492300608949964273411021602816.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 232
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[151, 191, 223, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226156437771606530900532845124795990839039906984601217899928546822043205632.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 246
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[152, 176, 208, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312848583677764512750242677738991171946451439834175447313734237415276544.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 158
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[153, 177, 209, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625697167355529025500485355477982343892902879668350894627468474830553088.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 55
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[154, 178, 210, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251394334711058051000970710955964687785805759336701789254936949661106176.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 95
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[155, 179, 211, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618502788669422116102001941421911929375571611518673403578509873899322212352.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 87
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[156, 180, 212, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005577338844232204003882843823858751143223037346807157019747798644424704.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 63
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[157, 181, 213, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011154677688464408007765687647717502286446074693614314039495597288849408.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 150
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[158, 182, 214, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022309355376928816015531375295435004572892149387228628078991194577698816.svg|420px]]
|-
|class=&quot;size&quot;| 4
|class=&quot;prop&quot;| 254
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[159, 183, 215, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044618710753857632031062750590870009145784298774457256157982389155397632.svg|420px]]
|}

[[Category:Boolf prop/3-ary|nameless 2]]</text>
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    </revision>
  </page>
  <page>
    <title>Template:Boolf prop/props.css</title>
    <ns>10</ns>
    <id>317257</id>
    <revision>
      <id>2692081</id>
      <timestamp>2024-12-15T21:42:14Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;table.boolf-props {text-align: center;}  table.boolf-props td.number-of-blocks {font-size: 80%; opacity: .5;}  table.boolf-props td.intpart span.count {font-size: 80%;}&quot;</comment>
      <origin>2692081</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="168" sha1="co6plwo8jej9ilwqq4ysuhptixj9k6c" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {font-size: 80%; opacity: .5;}

table.boolf-props td.intpart span.count {font-size: 80%;}</text>
      <sha1>co6plwo8jej9ilwqq4ysuhptixj9k6c</sha1>
    </revision>
    <revision>
      <id>2692083</id>
      <parentid>2692081</parentid>
      <timestamp>2024-12-15T21:51:00Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692083</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="291" sha1="286mo270tsld4ueig6wofjclssdx9zl" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {font-style: italic; opacity: .5; font-size: 70%;}

table.boolf-props td.intpart span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>286mo270tsld4ueig6wofjclssdx9zl</sha1>
    </revision>
    <revision>
      <id>2692098</id>
      <parentid>2692083</parentid>
      <timestamp>2024-12-15T23:11:35Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692098</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="318" sha1="iy7rdriucbrs4xio5s3lop9nuzjq23x" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {background-color: #f9f3ea; font-style: italic; opacity: .5; font-size: 70%;}

table.boolf-props td.intpart span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>iy7rdriucbrs4xio5s3lop9nuzjq23x</sha1>
    </revision>
    <revision>
      <id>2692099</id>
      <parentid>2692098</parentid>
      <timestamp>2024-12-15T23:13:15Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692099</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="375" sha1="gsitwmbt0x3tk0ixzqdc6y2xql2f0nq" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {background-color: #f9f3ea; font-style: italic; opacity: .5; font-size: 70%;}
table.boolf-props td.intpart {background-color: #f9f3ea;}
table.boolf-props td.intpart span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>gsitwmbt0x3tk0ixzqdc6y2xql2f0nq</sha1>
    </revision>
    <revision>
      <id>2692101</id>
      <parentid>2692099</parentid>
      <timestamp>2024-12-15T23:14:30Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692101</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="375" sha1="1miqqwlu1j5p1lnszdn3jm6jal6ykgx" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {background-color: #f9f3ea; font-style: italic; color: gray; font-size: 70%;}
table.boolf-props td.intpart {background-color: #f9f3ea;}
table.boolf-props td.intpart span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>1miqqwlu1j5p1lnszdn3jm6jal6ykgx</sha1>
    </revision>
    <revision>
      <id>2692102</id>
      <parentid>2692101</parentid>
      <timestamp>2024-12-15T23:14:57Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692102</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="375" sha1="jyr8fc51iqhl57zm1cc28b1369lg55a" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {background-color: #f9f3ea; font-style: italic; color: gray; font-size: 80%;}
table.boolf-props td.intpart {background-color: #f9f3ea;}
table.boolf-props td.intpart span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>jyr8fc51iqhl57zm1cc28b1369lg55a</sha1>
    </revision>
    <revision>
      <id>2692170</id>
      <parentid>2692102</parentid>
      <timestamp>2024-12-16T11:53:00Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692170</origin>
      <model>sanitized-css</model>
      <format>text/css</format>
      <text bytes="520" sha1="c86epv7i4u8e08uh6kl5s866c625dnv" xml:space="preserve">table.boolf-props {text-align: center;}

table.boolf-props td.number-of-blocks {background-color: #f9f3ea; font-style: italic; color: gray; font-size: 80%;}
table.boolf-props td.intpart {background-color: #f9f3ea;}
table.boolf-props td.intpart span.sortkey {display: none;}
table.boolf-props td.intpart span.formula {display: block;}
table.boolf-props td.intpart span.formula span.count {font-style: italic; padding-right: 2px;}
table.boolf-props td.intpart span.formula span.size {font-weight: bold; padding-left: 2px;}</text>
      <sha1>c86epv7i4u8e08uh6kl5s866c625dnv</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/great patron</title>
    <ns>0</ns>
    <id>317258</id>
    <revision>
      <id>2692104</id>
      <timestamp>2024-12-15T23:35:50Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| great_patron !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list...&quot;</comment>
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      <text bytes="2492" sha1="6ai6rk7mi88b1ttql431taf9a5dytoj" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great_patron
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 8, 9, 22, 23, 30, 31, 32, 33, 40, 41, 54, 55, 62, 63, 64, 65, 72, 73, 86, 87, 94, 95, 96, 97, 104, 105, 118, 119, 126, 127, 128, 129, 136, 137, 150, 151, 158, 159, 160, 161, 168, 169, 182, 183, 190, 191, 192, 193, 200, 201, 214, 215, 222, 223, 224, 225, 232, 233, 246, 247, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_87183322370842425795587566660030343945541236792690890543826997033971843334915.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 10, 11, 20, 21, 28, 29, 34, 35, 42, 43, 52, 53, 60, 61, 66, 67, 74, 75, 84, 85, 92, 93, 98, 99, 106, 107, 116, 117, 124, 125, 130, 131, 138, 139, 148, 149, 156, 157, 162, 163, 170, 171, 180, 181, 188, 189, 194, 195, 202, 203, 212, 213, 220, 221, 226, 227, 234, 235, 244, 245, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21795908540656425996854594658322324688933120046367589864458143256125197257740.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 12, 13, 18, 19, 26, 27, 36, 37, 44, 45, 50, 51, 58, 59, 68, 69, 76, 77, 82, 83, 90, 91, 100, 101, 108, 109, 114, 115, 122, 123, 132, 133, 140, 141, 146, 147, 154, 155, 164, 165, 172, 173, 178, 179, 186, 187, 196, 197, 204, 205, 210, 211, 218, 219, 228, 229, 236, 237, 242, 243, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5449288926947384691044460637839535982424523404371366380120111804560245010480.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 14, 15, 16, 17, 24, 25, 38, 39, 46, 47, 48, 49, 56, 57, 70, 71, 78, 79, 80, 81, 88, 89, 102, 103, 110, 111, 112, 113, 120, 121, 134, 135, 142, 143, 144, 145, 152, 153, 166, 167, 174, 175, 176, 177, 184, 185, 198, 199, 206, 207, 208, 209, 216, 217, 230, 231, 238, 239, 240, 241, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1363569398869958940084363052495703236371104422210717251052331913255844036800.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great_patron]]</text>
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    </revision>
    <revision>
      <id>2692110</id>
      <parentid>2692104</parentid>
      <timestamp>2024-12-16T00:03:16Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692110</origin>
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      <text bytes="2622" sha1="c7oz2mezzkf610vctw1nog68abt8w03" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great patron
!class=&quot;prop&quot;| patron tiling and slatting
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 8, 9, 22, 23, 30, 31, 32, 33, 40, 41, 54, 55, 62, 63, 64, 65, 72, 73, 86, 87, 94, 95, 96, 97, 104, 105, 118, 119, 126, 127, 128, 129, 136, 137, 150, 151, 158, 159, 160, 161, 168, 169, 182, 183, 190, 191, 192, 193, 200, 201, 214, 215, 222, 223, 224, 225, 232, 233, 246, 247, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_87183322370842425795587566660030343945541236792690890543826997033971843334915.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;prop&quot;| (0, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 10, 11, 20, 21, 28, 29, 34, 35, 42, 43, 52, 53, 60, 61, 66, 67, 74, 75, 84, 85, 92, 93, 98, 99, 106, 107, 116, 117, 124, 125, 130, 131, 138, 139, 148, 149, 156, 157, 162, 163, 170, 171, 180, 181, 188, 189, 194, 195, 202, 203, 212, 213, 220, 221, 226, 227, 234, 235, 244, 245, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21795908540656425996854594658322324688933120046367589864458143256125197257740.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;prop&quot;| (1, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 12, 13, 18, 19, 26, 27, 36, 37, 44, 45, 50, 51, 58, 59, 68, 69, 76, 77, 82, 83, 90, 91, 100, 101, 108, 109, 114, 115, 122, 123, 132, 133, 140, 141, 146, 147, 154, 155, 164, 165, 172, 173, 178, 179, 186, 187, 196, 197, 204, 205, 210, 211, 218, 219, 228, 229, 236, 237, 242, 243, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5449288926947384691044460637839535982424523404371366380120111804560245010480.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;prop&quot;| (1, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 14, 15, 16, 17, 24, 25, 38, 39, 46, 47, 48, 49, 56, 57, 70, 71, 78, 79, 80, 81, 88, 89, 102, 103, 110, 111, 112, 113, 120, 121, 134, 135, 142, 143, 144, 145, 152, 153, 166, 167, 174, 175, 176, 177, 184, 185, 198, 199, 206, 207, 208, 209, 216, 217, 230, 231, 238, 239, 240, 241, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1363569398869958940084363052495703236371104422210717251052331913255844036800.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great patron]]</text>
      <sha1>c7oz2mezzkf610vctw1nog68abt8w03</sha1>
    </revision>
  </page>
  <page>
    <title>File:LIB.2A.Shared.20241216.pdf</title>
    <ns>6</ns>
    <id>317259</id>
    <revision>
      <id>2692109</id>
      <timestamp>2024-12-15T23:47:08Z</timestamp>
      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
      </contributor>
      <comment>{{Information
|Description=LIB.2A: Shared Libraries (20241216 - 20241214-1)
|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}</comment>
      <origin>2692109</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="271" sha1="7rzx166aptju2tevvwfgrga7qwntfpx" xml:space="preserve">== Summary ==
{{Information
|Description=LIB.2A: Shared Libraries (20241216 - 20241214-1)
|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>7rzx166aptju2tevvwfgrga7qwntfpx</sha1>
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  </page>
  <page>
    <title>Boolf prop/3-ary/great quaestor</title>
    <ns>0</ns>
    <id>317260</id>
    <revision>
      <id>2692113</id>
      <timestamp>2024-12-16T00:05:38Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| great quaestor !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-li...&quot;</comment>
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      <text bytes="2496" sha1="1hjurdbofydkjh4paghyfeaa044sf6v" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great quaestor
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 14, 15, 22, 23, 24, 25, 36, 37, 42, 43, 50, 51, 60, 61, 66, 67, 76, 77, 84, 85, 90, 91, 102, 103, 104, 105, 112, 113, 126, 127, 128, 129, 142, 143, 150, 151, 152, 153, 164, 165, 170, 171, 178, 179, 188, 189, 194, 195, 204, 205, 212, 213, 218, 219, 230, 231, 232, 233, 240, 241, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86849393352013852949521188005354100301455703388471759192243584189472817856515.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 12, 13, 20, 21, 26, 27, 38, 39, 40, 41, 48, 49, 62, 63, 64, 65, 78, 79, 86, 87, 88, 89, 100, 101, 106, 107, 114, 115, 124, 125, 130, 131, 140, 141, 148, 149, 154, 155, 166, 167, 168, 169, 176, 177, 190, 191, 192, 193, 206, 207, 214, 215, 216, 217, 228, 229, 234, 235, 242, 243, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21732303012202958413271726050200185473229595476807730473600586671770529968140.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 10, 11, 18, 19, 28, 29, 32, 33, 46, 47, 54, 55, 56, 57, 70, 71, 72, 73, 80, 81, 94, 95, 98, 99, 108, 109, 116, 117, 122, 123, 132, 133, 138, 139, 146, 147, 156, 157, 160, 161, 174, 175, 182, 183, 184, 185, 198, 199, 200, 201, 208, 209, 222, 223, 226, 227, 236, 237, 244, 245, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5512894469673765216060020901188123495022080315085001639862785106052108782640.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 8, 9, 16, 17, 30, 31, 34, 35, 44, 45, 52, 53, 58, 59, 68, 69, 74, 75, 82, 83, 92, 93, 96, 97, 110, 111, 118, 119, 120, 121, 134, 135, 136, 137, 144, 145, 158, 159, 162, 163, 172, 173, 180, 181, 186, 187, 196, 197, 202, 203, 210, 211, 220, 221, 224, 225, 238, 239, 246, 247, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1697498403425618844718050051945498583562605485276072733750628040617673032640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great quaestor]]</text>
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    </revision>
    <revision>
      <id>2692114</id>
      <parentid>2692113</parentid>
      <timestamp>2024-12-16T00:08:18Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
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      <text bytes="2628" sha1="emlt4jm38zl2hrb60ch3b8mj7xgr8pq" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great quaestor
!class=&quot;prop&quot;| quaestor tiling and slatting
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 14, 15, 22, 23, 24, 25, 36, 37, 42, 43, 50, 51, 60, 61, 66, 67, 76, 77, 84, 85, 90, 91, 102, 103, 104, 105, 112, 113, 126, 127, 128, 129, 142, 143, 150, 151, 152, 153, 164, 165, 170, 171, 178, 179, 188, 189, 194, 195, 204, 205, 212, 213, 218, 219, 230, 231, 232, 233, 240, 241, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86849393352013852949521188005354100301455703388471759192243584189472817856515.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;prop&quot;| (0, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 12, 13, 20, 21, 26, 27, 38, 39, 40, 41, 48, 49, 62, 63, 64, 65, 78, 79, 86, 87, 88, 89, 100, 101, 106, 107, 114, 115, 124, 125, 130, 131, 140, 141, 148, 149, 154, 155, 166, 167, 168, 169, 176, 177, 190, 191, 192, 193, 206, 207, 214, 215, 216, 217, 228, 229, 234, 235, 242, 243, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21732303012202958413271726050200185473229595476807730473600586671770529968140.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;prop&quot;| (1, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 10, 11, 18, 19, 28, 29, 32, 33, 46, 47, 54, 55, 56, 57, 70, 71, 72, 73, 80, 81, 94, 95, 98, 99, 108, 109, 116, 117, 122, 123, 132, 133, 138, 139, 146, 147, 156, 157, 160, 161, 174, 175, 182, 183, 184, 185, 198, 199, 200, 201, 208, 209, 222, 223, 226, 227, 236, 237, 244, 245, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5512894469673765216060020901188123495022080315085001639862785106052108782640.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;prop&quot;| (1, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 8, 9, 16, 17, 30, 31, 34, 35, 44, 45, 52, 53, 58, 59, 68, 69, 74, 75, 82, 83, 92, 93, 96, 97, 110, 111, 118, 119, 120, 121, 134, 135, 136, 137, 144, 145, 158, 159, 162, 163, 172, 173, 180, 181, 186, 187, 196, 197, 202, 203, 210, 211, 220, 221, 224, 225, 238, 239, 246, 247, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1697498403425618844718050051945498583562605485276072733750628040617673032640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great quaestor]]</text>
      <sha1>emlt4jm38zl2hrb60ch3b8mj7xgr8pq</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/great prefect</title>
    <ns>0</ns>
    <id>317261</id>
    <revision>
      <id>2692116</id>
      <timestamp>2024-12-16T00:13:47Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| great prefect !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-lis...&quot;</comment>
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      <format>text/x-wiki</format>
      <text bytes="2475" sha1="8m789sjbvqj3ywe2nzg5qir3r9ovn7h" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great prefect
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 168, 169, 170, 171, 172, 173, 174, 175, 184, 185, 186, 187, 188, 189, 190, 191, 232, 233, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_115341536334051360635216133924802312858153080028147758503657919491080292598015.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31, 72, 73, 74, 75, 76, 77, 78, 79, 88, 89, 90, 91, 92, 93, 94, 95, 160, 161, 162, 163, 164, 165, 166, 167, 176, 177, 178, 179, 180, 181, 182, 183, 224, 225, 226, 227, 228, 229, 230, 231, 240, 241, 242, 243, 244, 245, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_450552876304888127481313023143759034602160468938870859140982054061458128640.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55, 96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119, 136, 137, 138, 139, 140, 141, 142, 143, 152, 153, 154, 155, 156, 157, 158, 159, 200, 201, 202, 203, 204, 205, 206, 207, 216, 217, 218, 219, 220, 221, 222, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26855044144683368651945175119863822971226346889019049307093482864640.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63, 104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151, 192, 193, 194, 195, 196, 197, 198, 199, 208, 209, 210, 211, 212, 213, 214, 215]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_104902516190169408796660840650921197327587787639633155677896048640.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great prefect]]</text>
      <sha1>8m789sjbvqj3ywe2nzg5qir3r9ovn7h</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/prefect</title>
    <ns>0</ns>
    <id>317262</id>
    <revision>
      <id>2692119</id>
      <timestamp>2024-12-16T00:17:34Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| prefect !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| (0, 0) |class=&quot;block&quot;| &lt;span class=&quot;block-li...&quot;</comment>
      <origin>2692119</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4561" sha1="79k2f24g7vxilh1lepzwkyihe9ncv1a" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| prefect
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_16711935.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (4, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4278255360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (2, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_71777214277877760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (6, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18374966855136706560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (1, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_308280787921469235485736960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (5, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[72, 73, 74, 75, 76, 77, 78, 79, 88, 89, 90, 91, 92, 93, 94, 95]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_78919881707896124284348661760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (3, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1324055902107822182681362917658460160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (7, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_338958310939602478766428906920565800960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (7, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5686776797628873740399791320050227284918927360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (3, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[136, 137, 138, 139, 140, 141, 142, 143, 152, 153, 154, 155, 156, 157, 158, 159]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1455814860192991677542346577932858184939245404160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (5, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[160, 161, 162, 163, 164, 165, 166, 167, 176, 177, 178, 179, 180, 181, 182, 183]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_24424520365467623060330297684840395266093667022599618560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (1, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[168, 169, 170, 171, 172, 173, 174, 175, 184, 185, 186, 187, 188, 189, 190, 191]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6252677213559711503444556207319141188119978757785502351360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (6, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[192, 193, 194, 195, 196, 197, 198, 199, 208, 209, 210, 211, 212, 213, 214, 215]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_104902516190169408790974063514334012647585517534779054617274613760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (2, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[200, 201, 202, 203, 204, 205, 206, 207, 216, 217, 218, 219, 220, 221, 222, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26855044144683368650489360259669507237781892488903437982022301122560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (4, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[224, 225, 226, 227, 228, 229, 230, 231, 240, 241, 242, 243, 244, 245, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_450552876304888127456888502778291411541830171175110582166992262750231592960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[232, 233, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_115341536334051360628963456711242601354708523820828309034750019264059287797760.svg|420px]]
|}

[[Category:Boolf prop/3-ary|prefect]]</text>
      <sha1>79k2f24g7vxilh1lepzwkyihe9ncv1a</sha1>
    </revision>
    <revision>
      <id>2692120</id>
      <parentid>2692119</parentid>
      <timestamp>2024-12-16T00:21:13Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692120</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="4633" sha1="4aaqil56l8rp8a3w5b6cy16x84v25dx" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;source lang=&quot;python&quot;&gt;
val = boolf.prefect_walsh_and_oddness
&lt;/source&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| prefect
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_16711935.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (4, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 9, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, 31]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4278255360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (2, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[32, 33, 34, 35, 36, 37, 38, 39, 48, 49, 50, 51, 52, 53, 54, 55]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_71777214277877760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (6, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[40, 41, 42, 43, 44, 45, 46, 47, 56, 57, 58, 59, 60, 61, 62, 63]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18374966855136706560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (1, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_308280787921469235485736960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (5, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[72, 73, 74, 75, 76, 77, 78, 79, 88, 89, 90, 91, 92, 93, 94, 95]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_78919881707896124284348661760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (3, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[96, 97, 98, 99, 100, 101, 102, 103, 112, 113, 114, 115, 116, 117, 118, 119]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1324055902107822182681362917658460160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (7, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[104, 105, 106, 107, 108, 109, 110, 111, 120, 121, 122, 123, 124, 125, 126, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_338958310939602478766428906920565800960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (7, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[128, 129, 130, 131, 132, 133, 134, 135, 144, 145, 146, 147, 148, 149, 150, 151]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5686776797628873740399791320050227284918927360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (3, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[136, 137, 138, 139, 140, 141, 142, 143, 152, 153, 154, 155, 156, 157, 158, 159]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1455814860192991677542346577932858184939245404160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (5, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[160, 161, 162, 163, 164, 165, 166, 167, 176, 177, 178, 179, 180, 181, 182, 183]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_24424520365467623060330297684840395266093667022599618560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (1, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[168, 169, 170, 171, 172, 173, 174, 175, 184, 185, 186, 187, 188, 189, 190, 191]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_6252677213559711503444556207319141188119978757785502351360.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (6, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[192, 193, 194, 195, 196, 197, 198, 199, 208, 209, 210, 211, 212, 213, 214, 215]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_104902516190169408790974063514334012647585517534779054617274613760.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (2, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[200, 201, 202, 203, 204, 205, 206, 207, 216, 217, 218, 219, 220, 221, 222, 223]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26855044144683368650489360259669507237781892488903437982022301122560.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (4, 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[224, 225, 226, 227, 228, 229, 230, 231, 240, 241, 242, 243, 244, 245, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_450552876304888127456888502778291411541830171175110582166992262750231592960.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| (0, 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[232, 233, 234, 235, 236, 237, 238, 239, 248, 249, 250, 251, 252, 253, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_115341536334051360628963456711242601354708523820828309034750019264059287797760.svg|420px]]
|}

[[Category:Boolf prop/3-ary|prefect]]</text>
      <sha1>4aaqil56l8rp8a3w5b6cy16x84v25dx</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/praetor</title>
    <ns>0</ns>
    <id>317263</id>
    <revision>
      <id>2692122</id>
      <timestamp>2024-12-16T00:23:08Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| praetor !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[...&quot;</comment>
      <origin>2692122</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5004" sha1="sbadlnoyo49s9qesev7jzdoigo3mw4c" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| praetor
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896486333794311352466587372716853682903935386526539088661048023915243372545.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 16, 35, 50, 69, 84, 103, 118, 137, 152, 171, 186, 205, 220, 239, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948905734546486136869951883554333513920930453733235058270336113397551136770.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 19, 32, 49, 70, 87, 100, 117, 138, 155, 168, 185, 206, 223, 236, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474121596086149254626164786388727017944227235188016128539048712578749956100.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 18, 33, 48, 71, 86, 101, 116, 139, 154, 169, 184, 207, 222, 237, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237226439955551866896230251618139443684273378497113829943559501384618606600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 21, 38, 55, 64, 81, 98, 115, 140, 157, 174, 191, 200, 217, 234, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618530395862144462655476466049913335740215903444871749035668935404792315920.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 20, 39, 54, 65, 80, 99, 114, 141, 156, 175, 190, 201, 216, 235, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809306608409155385117565147909784777930941306599910009599991384103204159520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 23, 36, 53, 66, 83, 96, 113, 142, 159, 172, 189, 202, 219, 232, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904632599755384329195713988645480943156445146135475740806812471150338113600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 22, 37, 52, 67, 82, 97, 112, 143, 158, 173, 188, 203, 218, 233, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452326652497221992071812679864255262735934189248943787820503656239277801600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 25, 42, 59, 76, 93, 110, 127, 128, 145, 162, 179, 196, 213, 230, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226158149741384028720572606924675209868983382979970295467139156671320817920.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 24, 43, 58, 77, 92, 111, 126, 129, 144, 163, 178, 197, 212, 231, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113081663025572211472148249545134115373825524354682453474974754906998571520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 27, 40, 57, 78, 95, 108, 125, 130, 147, 160, 177, 198, 215, 228, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56539537484711520525883456196830964956379609018678856372368873916597273600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 26, 41, 56, 79, 94, 109, 124, 131, 146, 161, 176, 199, 214, 229, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28270415781076374480063399420383357223159665345770485827991357936004761600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 29, 46, 63, 72, 89, 106, 123, 132, 149, 166, 183, 192, 209, 226, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14134884358836501807247954945507473978368961208853467913361413538603601920.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 28, 47, 62, 73, 88, 107, 122, 133, 148, 167, 182, 193, 208, 227, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7067603939098263223115488859022596794109732589517586231381598960068075520.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 31, 44, 61, 74, 91, 104, 121, 134, 151, 164, 181, 194, 211, 224, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3533721092794470035920757768146409936863299563418306387964880433987993600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1766900986317273406530518280719747121391491205567195105433173375773081600.svg|420px]]
|}

[[Category:Boolf prop/3-ary|praetor]]</text>
      <sha1>sbadlnoyo49s9qesev7jzdoigo3mw4c</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/great praetor</title>
    <ns>0</ns>
    <id>317264</id>
    <revision>
      <id>2692123</id>
      <timestamp>2024-12-16T00:27:04Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| great praetor !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-lis...&quot;</comment>
      <origin>2692123</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="2494" sha1="tot409vncq4edwyusfeffpv3ezul0qr" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great praetor
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 8, 9, 16, 17, 24, 25, 34, 35, 42, 43, 50, 51, 58, 59, 68, 69, 76, 77, 84, 85, 92, 93, 102, 103, 110, 111, 118, 119, 126, 127, 128, 129, 136, 137, 144, 145, 152, 153, 162, 163, 170, 171, 178, 179, 186, 187, 196, 197, 204, 205, 212, 213, 220, 221, 230, 231, 238, 239, 246, 247, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_87184631881107753729529260112740996522067674747594426895873498048891113898755.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 10, 11, 18, 19, 26, 27, 32, 33, 40, 41, 48, 49, 56, 57, 70, 71, 78, 79, 86, 87, 94, 95, 100, 101, 108, 109, 116, 117, 124, 125, 130, 131, 138, 139, 146, 147, 154, 155, 160, 161, 168, 169, 176, 177, 184, 185, 198, 199, 206, 207, 214, 215, 222, 223, 228, 229, 236, 237, 244, 245, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_21796157989307489016528341893624080783808039888049579300682968445815970597900.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[4, 5, 12, 13, 20, 21, 28, 29, 38, 39, 46, 47, 54, 55, 62, 63, 64, 65, 72, 73, 80, 81, 88, 89, 98, 99, 106, 107, 114, 115, 122, 123, 132, 133, 140, 141, 148, 149, 156, 157, 166, 167, 174, 175, 182, 183, 190, 191, 192, 193, 200, 201, 208, 209, 216, 217, 226, 227, 234, 235, 242, 243, 250, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5449039492569234612803405057764228184443635903843152812780403332006668152880.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[6, 7, 14, 15, 22, 23, 30, 31, 36, 37, 44, 45, 52, 53, 60, 61, 66, 67, 74, 75, 82, 83, 90, 91, 96, 97, 104, 105, 112, 113, 120, 121, 134, 135, 142, 143, 150, 151, 158, 159, 164, 165, 172, 173, 180, 181, 188, 189, 194, 195, 202, 203, 210, 211, 218, 219, 224, 225, 232, 233, 240, 241, 248, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1362259874331718064709977944558602362950634126153405030120714181199376990400.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great praetor]]</text>
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  </page>
  <page>
    <title>Boolf prop/3-ary/quaestor</title>
    <ns>0</ns>
    <id>317265</id>
    <revision>
      <id>2692125</id>
      <timestamp>2024-12-16T00:29:09Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| quaestor !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;...&quot;</comment>
      <origin>2692125</origin>
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      <text bytes="5006" sha1="ffl2a6z5zv2t67arc7v48xzd0ue2c8q" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| quaestor
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 24, 36, 60, 66, 90, 102, 126, 129, 153, 165, 189, 195, 219, 231, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896048070373769491417295362101480869679047472698551317454488802455720034305.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 25, 37, 61, 67, 91, 103, 127, 128, 152, 164, 188, 194, 218, 230, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948024035186884745708647681050740434967129631552388094066724205209612976130.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 26, 38, 62, 64, 88, 100, 124, 131, 155, 167, 191, 193, 217, 229, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474012017593442375796715454410890738509434893084198442928681539648131956740.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 27, 39, 63, 65, 89, 101, 125, 130, 154, 166, 190, 192, 216, 228, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237006008796721187898357727205445369286618920342877330305695904669559160840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 28, 32, 56, 70, 94, 98, 122, 133, 157, 161, 185, 199, 223, 227, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618503017825678581285093526099662564752582377139826828513203328849400561680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 29, 33, 57, 71, 95, 99, 123, 132, 156, 160, 184, 198, 222, 226, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251508912839290642546763049831282384266557049702447293455176419243982880.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 30, 34, 58, 68, 92, 96, 120, 135, 159, 163, 187, 197, 221, 225, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904625754456419645505172858075163517505663897491222935475652947564535218240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 31, 35, 59, 69, 93, 97, 121, 134, 158, 162, 186, 196, 220, 224, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452312877228209822752586429037581758754825790865558725997445175773036019840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 16, 44, 52, 74, 82, 110, 118, 137, 145, 173, 181, 203, 211, 239, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_227039847827326250973527176555168871534410259539694997583804615943822311680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 17, 45, 53, 75, 83, 111, 119, 136, 144, 172, 180, 202, 210, 238, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113519923913663125486763588277584435767705537379596074693725301336279482880.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 18, 46, 54, 72, 80, 108, 116, 139, 147, 175, 183, 201, 209, 237, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56759961956831562754920408025753098590070852662023480053763613825614218240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 19, 47, 55, 73, 81, 109, 117, 138, 146, 174, 182, 200, 208, 236, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28379980978415781377460204012876549295160528233448884002362986957850019840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 20, 40, 48, 78, 86, 106, 114, 141, 149, 169, 177, 207, 215, 235, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14189990541863233051101912389232910289006925269952906176366478028606279680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 21, 41, 49, 79, 87, 107, 115, 140, 148, 168, 176, 206, 214, 234, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7094995270931616525550956194616455144534738110701794189842749424232570880.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 22, 42, 50, 76, 84, 104, 112, 143, 151, 171, 179, 205, 213, 233, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3547497635465808263496641467919331206345643567925630297969809815666442240.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 23, 43, 51, 77, 85, 105, 113, 142, 150, 170, 178, 204, 212, 232, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1773748817732904131748320733959665603180640652894150424401371991818403840.svg|420px]]
|}

[[Category:Boolf prop/3-ary|quaestor]]</text>
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    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/patron</title>
    <ns>0</ns>
    <id>317266</id>
    <revision>
      <id>2692128</id>
      <timestamp>2024-12-16T00:37:28Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| patron !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 16 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0...&quot;</comment>
      <origin>2692128</origin>
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      <text bytes="5020" sha1="4jgwyf4ysbelut747ctcc0tbbp96j3s" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| patron
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 254
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044672630650405809399918354010442245591851645859933116841437057859452930.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 168
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 28, 42, 52, 74, 84, 98, 124, 130, 156, 170, 180, 202, 212, 226, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005685178637328558741594050663002905020638742847807184697897668403265540.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 86
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 29, 43, 53, 75, 85, 99, 125, 131, 157, 171, 181, 203, 213, 227, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011370357274657117483188101326005810041277485695614369395795336806531080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 200
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 26, 44, 50, 76, 82, 100, 122, 132, 154, 172, 178, 204, 210, 228, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251825693883443469951815538312541223541788834173213976545346561784152080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 54
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 27, 45, 51, 77, 83, 101, 123, 133, 155, 173, 179, 205, 211, 229, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618503651387766886939903631076625082447083577668346427953090693123568304160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 96
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 24, 46, 48, 78, 80, 102, 120, 134, 152, 174, 176, 206, 208, 230, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452314574020367306188553621987165297295031963802388352215619825219179380800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 158
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 25, 47, 49, 79, 81, 103, 121, 135, 153, 175, 177, 207, 209, 231, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904629148040734612377107243974330594590063927604776704431239650438358761600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 128
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113085120632150062291155594166442760724283005074033548050578292795018051840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 126
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 23, 33, 63, 65, 95, 105, 119, 137, 151, 161, 191, 193, 223, 233, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226170241264300124582311188332885521448566010148067096101156585590036103680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 40
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 20, 34, 60, 66, 92, 106, 116, 138, 148, 162, 188, 194, 220, 234, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28297161706838003726123292056778560072686043379682147634683187706662487040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 214
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 21, 35, 61, 67, 93, 107, 117, 139, 149, 163, 189, 195, 221, 235, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56594323413676007452246584113557120145372086759364295269366375413324974080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 72
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 18, 36, 58, 68, 90, 108, 114, 140, 146, 164, 186, 196, 218, 236, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7177816621911453544868397074866119584632679289615579396825254958297518080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 182
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 19, 37, 59, 69, 91, 109, 115, 141, 147, 165, 187, 197, 219, 237, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14355633243822907089736794149732239169265358579231158793650509916595036160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 224
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 16, 38, 56, 70, 88, 110, 112, 142, 144, 166, 184, 198, 216, 238, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2208558936285673839567395511402448162002843601184064801824145866101964800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 30
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 17, 39, 57, 71, 89, 111, 113, 143, 145, 167, 185, 199, 217, 239, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4417117872571347679134791022804896324005687202368129603648291732203929600.svg|420px]]
|}

[[Category:Boolf prop/3-ary|patron]]</text>
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    </revision>
    <revision>
      <id>2692131</id>
      <parentid>2692128</parentid>
      <timestamp>2024-12-16T00:40:30Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <origin>2692131</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="5326" sha1="47c4hybc947bs0fhc7vk657i22z8e79" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| patron
!class=&quot;prop&quot;| patron index
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 0
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 30, 40, 54, 72, 86, 96, 126, 128, 158, 168, 182, 200, 214, 224, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28948022336315325202904699959177005221122795925822929966558420718528929726465.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 254
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 31, 41, 55, 73, 87, 97, 127, 129, 159, 169, 183, 201, 215, 225, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_57896044672630650405809399918354010442245591851645859933116841437057859452930.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 168
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 28, 42, 52, 74, 84, 98, 124, 130, 156, 170, 180, 202, 212, 226, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005685178637328558741594050663002905020638742847807184697897668403265540.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 86
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 29, 43, 53, 75, 85, 99, 125, 131, 157, 171, 181, 203, 213, 227, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011370357274657117483188101326005810041277485695614369395795336806531080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 200
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 26, 44, 50, 76, 82, 100, 122, 132, 154, 172, 178, 204, 210, 228, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251825693883443469951815538312541223541788834173213976545346561784152080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 54
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 27, 45, 51, 77, 83, 101, 123, 133, 155, 173, 179, 205, 211, 229, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618503651387766886939903631076625082447083577668346427953090693123568304160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 96
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 24, 46, 48, 78, 80, 102, 120, 134, 152, 174, 176, 206, 208, 230, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452314574020367306188553621987165297295031963802388352215619825219179380800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 158
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 25, 47, 49, 79, 81, 103, 121, 135, 153, 175, 177, 207, 209, 231, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904629148040734612377107243974330594590063927604776704431239650438358761600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 128
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113085120632150062291155594166442760724283005074033548050578292795018051840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 126
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 23, 33, 63, 65, 95, 105, 119, 137, 151, 161, 191, 193, 223, 233, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226170241264300124582311188332885521448566010148067096101156585590036103680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 40
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 20, 34, 60, 66, 92, 106, 116, 138, 148, 162, 188, 194, 220, 234, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28297161706838003726123292056778560072686043379682147634683187706662487040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 214
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 21, 35, 61, 67, 93, 107, 117, 139, 149, 163, 189, 195, 221, 235, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56594323413676007452246584113557120145372086759364295269366375413324974080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 72
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 18, 36, 58, 68, 90, 108, 114, 140, 146, 164, 186, 196, 218, 236, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7177816621911453544868397074866119584632679289615579396825254958297518080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 182
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 19, 37, 59, 69, 91, 109, 115, 141, 147, 165, 187, 197, 219, 237, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14355633243822907089736794149732239169265358579231158793650509916595036160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 224
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 16, 38, 56, 70, 88, 110, 112, 142, 144, 166, 184, 198, 216, 238, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2208558936285673839567395511402448162002843601184064801824145866101964800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 30
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 17, 39, 57, 71, 89, 111, 113, 143, 145, 167, 185, 199, 217, 239, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4417117872571347679134791022804896324005687202368129603648291732203929600.svg|420px]]
|}

[[Category:Boolf prop/3-ary|patron]]</text>
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    <revision>
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      <timestamp>2024-12-16T00:42:07Z</timestamp>
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      <text bytes="5522" sha1="8kdigw9w8wn186ix86bo3ja4d3rkhoz" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

[[File:Set of 3-ary Boolean functions 1809251421294659332139685398512665750726255159685711951796174474417100816385.svg|thumb|500px|similar pattern to [[Boolf prop/3-ary/nameless 1|nameless 1]]]]

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;16&lt;/span&gt;
&lt;/div&gt;

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!class=&quot;prop&quot;| patron
!class=&quot;prop&quot;| patron index
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|-
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|-
|class=&quot;size&quot;| 16
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|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 168
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 28, 42, 52, 74, 84, 98, 124, 130, 156, 170, 180, 202, 212, 226, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7237005685178637328558741594050663002905020638742847807184697897668403265540.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 86
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 29, 43, 53, 75, 85, 99, 125, 131, 157, 171, 181, 203, 213, 227, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14474011370357274657117483188101326005810041277485695614369395795336806531080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 200
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[4, 26, 44, 50, 76, 82, 100, 122, 132, 154, 172, 178, 204, 210, 228, 250]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1809251825693883443469951815538312541223541788834173213976545346561784152080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 54
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[5, 27, 45, 51, 77, 83, 101, 123, 133, 155, 173, 179, 205, 211, 229, 251]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3618503651387766886939903631076625082447083577668346427953090693123568304160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 96
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[6, 24, 46, 48, 78, 80, 102, 120, 134, 152, 174, 176, 206, 208, 230, 248]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_452314574020367306188553621987165297295031963802388352215619825219179380800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 158
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[7, 25, 47, 49, 79, 81, 103, 121, 135, 153, 175, 177, 207, 209, 231, 249]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_904629148040734612377107243974330594590063927604776704431239650438358761600.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 128
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 22, 32, 62, 64, 94, 104, 118, 136, 150, 160, 190, 192, 222, 232, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113085120632150062291155594166442760724283005074033548050578292795018051840.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 126
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 23, 33, 63, 65, 95, 105, 119, 137, 151, 161, 191, 193, 223, 233, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226170241264300124582311188332885521448566010148067096101156585590036103680.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 40
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[10, 20, 34, 60, 66, 92, 106, 116, 138, 148, 162, 188, 194, 220, 234, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_28297161706838003726123292056778560072686043379682147634683187706662487040.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 214
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[11, 21, 35, 61, 67, 93, 107, 117, 139, 149, 163, 189, 195, 221, 235, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_56594323413676007452246584113557120145372086759364295269366375413324974080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 72
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[12, 18, 36, 58, 68, 90, 108, 114, 140, 146, 164, 186, 196, 218, 236, 242]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_7177816621911453544868397074866119584632679289615579396825254958297518080.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 182
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[13, 19, 37, 59, 69, 91, 109, 115, 141, 147, 165, 187, 197, 219, 237, 243]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_14355633243822907089736794149732239169265358579231158793650509916595036160.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 224
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[14, 16, 38, 56, 70, 88, 110, 112, 142, 144, 166, 184, 198, 216, 238, 240]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_2208558936285673839567395511402448162002843601184064801824145866101964800.svg|420px]]
|-
|class=&quot;size&quot;| 16
|class=&quot;prop&quot;| 30
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[15, 17, 39, 57, 71, 89, 111, 113, 143, 145, 167, 185, 199, 217, 239, 241]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_4417117872571347679134791022804896324005687202368129603648291732203929600.svg|420px]]
|}

[[Category:Boolf prop/3-ary|patron]]</text>
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      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| quadrant !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 64 |class=&quot;prop&quot;| 0 |class=&quot;block&quot;| &lt;span class=&quot;block-list sma...&quot;</comment>
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&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;64&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
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!class=&quot;prop&quot;| quadrant
!class=&quot;block&quot;| block
|-
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|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_113427455640312821154458202477256070485.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_226854911280625642308916404954512140970.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192, 194, 196, 198, 200, 202, 204, 206, 208, 210, 212, 214, 216, 218, 220, 222, 224, 226, 228, 230, 232, 234, 236, 238, 240, 242, 244, 246, 248, 250, 252, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_38597363079105398474523661669562635950976567432906541858664736466827120476160.svg|420px]]
|-
|class=&quot;size&quot;| 64
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[129, 131, 133, 135, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 173, 175, 177, 179, 181, 183, 185, 187, 189, 191, 193, 195, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249, 251, 253, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_77194726158210796949047323339125271901953134865813083717329472933654240952320.svg|420px]]
|}

[[Category:Boolf prop/3-ary|quadrant]]</text>
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      <comment>New resource with &quot;A Student Studying Mechatronics&quot;</comment>
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      <comment>{{Information
|Description=LCal.9A: Recursion (20241216 - 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-16
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|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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{{Information
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|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Description=ARM.2ASM: Branch and Return Methods (20241216 - 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
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|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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|Description=Work2.1A: Libraries (20241216 - 20241214)
|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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|Source={{own|Young1lim}}
|Date=2024-12-16
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
}}

== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
      <sha1>6tger1lshc6qyrytw7nae8s2w8bw74e</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/guild</title>
    <ns>0</ns>
    <id>317272</id>
    <revision>
      <id>2692150</id>
      <timestamp>2024-12-16T10:34:32Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;20&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| guild !class=&quot;block&quot;| block |- |class=&quot;size&quot;| 10...&quot;</comment>
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      <text bytes="5635" sha1="6zkr79f65xu1kvzuhtgigupa7ehiwwd" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;20&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;16&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;10&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;24&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| guild
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[0, 6, 18, 20, 40, 60, 72, 90, 96, 102]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5151068508189986729329020895297.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 1
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[1, 7, 19, 21, 41, 61, 73, 91, 97, 103]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_10302137016379973458658041790594.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[2, 4, 16, 22, 42, 62, 76, 94, 112, 118]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_337499315611879986895982990586871828.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 3
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[3, 5, 17, 23, 43, 63, 77, 95, 113, 119]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_674998631223759973791965981173743656.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[8, 14, 32, 50, 64, 84, 104, 110, 122, 124]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_26585878272341897648215401821647159552.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 5
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[9, 15, 33, 51, 65, 85, 105, 111, 123, 125]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_53171756544683795296430803643294319104.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| guildless&lt;br&gt;(quadrant 0)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[10, 12, 26, 28, 34, 38, 44, 46, 48, 52, 56, 58, 68, 70, 74, 78, 80, 82, 88, 92, 98, 100, 114, 116]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_103847527001559230943823442414146560.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| guildless&lt;br&gt;(quadrant 1)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[11, 13, 27, 29, 35, 39, 45, 47, 49, 53, 57, 59, 69, 71, 75, 79, 81, 83, 89, 93, 99, 101, 115, 117]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_207695054003118461887646884828293120.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[24, 30, 36, 54, 66, 86, 106, 108, 120, 126]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_86400225374288976098416264893586997248.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 7
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[25, 31, 37, 55, 67, 87, 107, 109, 121, 127]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_172800450748577952196832529787173994496.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 8
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[128, 134, 146, 148, 168, 188, 200, 218, 224, 230]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_1752817784138796178891744461050445976218270197534802054197037531922432.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 9
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[129, 135, 147, 149, 169, 189, 201, 219, 225, 231]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_3505635568277592357783488922100891952436540395069604108394075063844864.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 10
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[130, 132, 144, 150, 170, 190, 204, 222, 240, 246]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_114845065950607360788559360485248519724259005605996089615563328969473261568.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 11
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[131, 133, 145, 151, 171, 191, 205, 223, 241, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_229690131901214721577118720970497039448518011211992179231126657938946523136.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 12
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[136, 142, 160, 178, 192, 212, 232, 238, 250, 252]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_9046705585184451178619436147836815220101223768690461940839254447669706227712.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 13
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[137, 143, 161, 179, 193, 213, 233, 239, 251, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_18093411170368902357238872295673630440202447537380923881678508895339412455424.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| guildless&lt;br&gt;(quadrant 2)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[138, 140, 154, 156, 162, 166, 172, 174, 176, 180, 184, 186, 196, 198, 202, 206, 208, 210, 216, 220, 226, 228, 242, 244]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_35337482286976642745991917876924801731222424797041072021802991043854991360.svg|420px]]
|-
|class=&quot;size&quot;| 24
|class=&quot;prop&quot;| guildless&lt;br&gt;(quadrant 3)
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[139, 141, 155, 157, 163, 167, 173, 175, 177, 181, 185, 187, 197, 199, 203, 207, 209, 211, 217, 221, 227, 229, 243, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_70674964573953285491983835753849603462444849594082144043605982087709982720.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 14
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[152, 158, 164, 182, 194, 214, 234, 236, 248, 254]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_29400473192865579153573495351619186358973886015542845221386061502106554073088.svg|420px]]
|-
|class=&quot;size&quot;| 10
|class=&quot;prop&quot;| 15
|class=&quot;block&quot;| &lt;span class=&quot;block-list&quot;&gt;[153, 159, 165, 183, 195, 215, 235, 237, 249, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_58800946385731158307146990703238372717947772031085690442772123004213108146176.svg|420px]]
|}

[[Category:Boolf prop/3-ary|guild]]</text>
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    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/great guild</title>
    <ns>0</ns>
    <id>317273</id>
    <revision>
      <id>2692153</id>
      <timestamp>2024-12-16T10:46:42Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| great guild !class=&quot;block&quot;| block |- |class=&quot;size&quot;...&quot;</comment>
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      <text bytes="2771" sha1="t34u58y4gwr1h3qe7pjv1e6lthyk04t" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;5&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;4&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;40&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| great guild
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 40
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 6, 7, 18, 19, 20, 21, 40, 41, 60, 61, 72, 73, 90, 91, 96, 97, 102, 103, 128, 129, 134, 135, 146, 147, 148, 149, 168, 169, 188, 189, 200, 201, 218, 219, 224, 225, 230, 231]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_5258453352416388536675233383151337928670263798128976122779099658453187.svg|420px]]
|-
|class=&quot;size&quot;| 40
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 4, 5, 16, 17, 22, 23, 42, 43, 62, 63, 76, 77, 94, 95, 112, 113, 118, 119, 130, 131, 132, 133, 144, 145, 150, 151, 170, 171, 190, 191, 204, 205, 222, 223, 240, 241, 246, 247]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_344535197851822082365678081455745559173789514764823908807377935880180400188.svg|420px]]
|-
|class=&quot;size&quot;| 40
|class=&quot;prop&quot;| 4
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[8, 9, 14, 15, 32, 33, 50, 51, 64, 65, 84, 85, 104, 105, 110, 111, 122, 123, 124, 125, 136, 137, 142, 143, 160, 161, 178, 179, 192, 193, 212, 213, 232, 233, 238, 239, 250, 251, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_27140116755553353535858308443510445660383428940888411515462409548474060161792.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| guildless
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[10, 11, 12, 13, 26, 27, 28, 29, 34, 35, 38, 39, 44, 45, 46, 47, 48, 49, 52, 53, 56, 57, 58, 59, 68, 69, 70, 71, 74, 75, 78, 79, 80, 81, 82, 83, 88, 89, 92, 93, 98, 99, 100, 101, 114, 115, 116, 117, 138, 139, 140, 141, 154, 155, 156, 157, 162, 163, 166, 167, 172, 173, 174, 175, 176, 177, 180, 181, 184, 185, 186, 187, 196, 197, 198, 199, 202, 203, 206, 207, 208, 209, 210, 211, 216, 217, 220, 221, 226, 227, 228, 229, 242, 243, 244, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_106012446860929928237975753630774405193978816972127893758240443458807413760.svg|420px]]
|-
|class=&quot;size&quot;| 40
|class=&quot;prop&quot;| 6
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[24, 25, 30, 31, 36, 37, 54, 55, 66, 67, 86, 87, 106, 107, 108, 109, 120, 121, 126, 127, 152, 153, 158, 159, 164, 165, 182, 183, 194, 195, 214, 215, 234, 235, 236, 237, 248, 249, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_88201419578596737460720486054857559077180858722751402592453433301000423211008.svg|420px]]
|}

[[Category:Boolf prop/3-ary|great guild]]</text>
      <sha1>t34u58y4gwr1h3qe7pjv1e6lthyk04t</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/greater guild</title>
    <ns>0</ns>
    <id>317274</id>
    <revision>
      <id>2692155</id>
      <timestamp>2024-12-16T10:49:21Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt; &lt;/div&gt;  {| class=&quot;wikitable sortable boolf-blocks&quot; !class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt; !class=&quot;prop&quot;| greater guild !class=&quot;block&quot;| block |- |class=&quot;siz...&quot;</comment>
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      <format>text/x-wiki</format>
      <text bytes="2344" sha1="e9hijlm5dsxwhv5nzuuipxvzeoa3hvb" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;3&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;2&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;80&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;1&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;96&lt;/span&gt;
&lt;/div&gt;

{| class=&quot;wikitable sortable boolf-blocks&quot;
!class=&quot;size&quot;| &lt;abbr title=&quot;block size&quot;&gt;#&lt;/abbr&gt;
!class=&quot;prop&quot;| greater guild
!class=&quot;block&quot;| block
|-
|class=&quot;size&quot;| 80
|class=&quot;prop&quot;| 0
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[0, 1, 6, 7, 18, 19, 20, 21, 24, 25, 30, 31, 36, 37, 40, 41, 54, 55, 60, 61, 66, 67, 72, 73, 86, 87, 90, 91, 96, 97, 102, 103, 106, 107, 108, 109, 120, 121, 126, 127, 128, 129, 134, 135, 146, 147, 148, 149, 152, 153, 158, 159, 164, 165, 168, 169, 182, 183, 188, 189, 194, 195, 200, 201, 214, 215, 218, 219, 224, 225, 230, 231, 234, 235, 236, 237, 248, 249, 254, 255]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_88201424837050089877109022730090942228518787393015200721429556080100081664195.svg|420px]]
|-
|class=&quot;size&quot;| 80
|class=&quot;prop&quot;| 2
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[2, 3, 4, 5, 8, 9, 14, 15, 16, 17, 22, 23, 32, 33, 42, 43, 50, 51, 62, 63, 64, 65, 76, 77, 84, 85, 94, 95, 104, 105, 110, 111, 112, 113, 118, 119, 122, 123, 124, 125, 130, 131, 132, 133, 136, 137, 142, 143, 144, 145, 150, 151, 160, 161, 170, 171, 178, 179, 190, 191, 192, 193, 204, 205, 212, 213, 222, 223, 232, 233, 238, 239, 240, 241, 246, 247, 250, 251, 252, 253]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_27484651953405175618223986524966191219557218455653235424269787484354240561980.svg|420px]]
|-
|class=&quot;size&quot;| 96
|class=&quot;prop&quot;| guildless
|class=&quot;block&quot;| &lt;span class=&quot;block-list small&quot;&gt;[10, 11, 12, 13, 26, 27, 28, 29, 34, 35, 38, 39, 44, 45, 46, 47, 48, 49, 52, 53, 56, 57, 58, 59, 68, 69, 70, 71, 74, 75, 78, 79, 80, 81, 82, 83, 88, 89, 92, 93, 98, 99, 100, 101, 114, 115, 116, 117, 138, 139, 140, 141, 154, 155, 156, 157, 162, 163, 166, 167, 172, 173, 174, 175, 176, 177, 180, 181, 184, 185, 186, 187, 196, 197, 198, 199, 202, 203, 206, 207, 208, 209, 210, 211, 216, 217, 220, 221, 226, 227, 228, 229, 242, 243, 244, 245]&lt;/span&gt;[[File:Set_of_3-ary_Boolean_functions_106012446860929928237975753630774405193978816972127893758240443458807413760.svg|420px]]
|}

[[Category:Boolf prop/3-ary|greater guild]]</text>
      <sha1>e9hijlm5dsxwhv5nzuuipxvzeoa3hvb</sha1>
    </revision>
  </page>
  <page>
    <title>Boolf prop/3-ary/principality</title>
    <ns>0</ns>
    <id>317275</id>
    <revision>
      <id>2692159</id>
      <timestamp>2024-12-16T11:23:11Z</timestamp>
      <contributor>
        <username>Watchduck</username>
        <id>137431</id>
      </contributor>
      <comment>New resource with &quot;&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;  &lt;div class=&quot;intpart&quot;&gt; &lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;44&lt;/span&gt;&lt;/span&gt; Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span clas...&quot;</comment>
      <origin>2692159</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="10174" sha1="ak59m6jgeyi9j3zvei7o4xf00ey5pqs" xml:space="preserve">&lt;templatestyles src=&quot;Boolf prop/blocks.css&quot; /&gt;

&lt;div class=&quot;intpart&quot;&gt;
&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;44&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;
&lt;/div&gt;

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[[Category:Boolf prop/3-ary|principality]]</text>
      <sha1>ak59m6jgeyi9j3zvei7o4xf00ey5pqs</sha1>
    </revision>
  </page>
  <page>
    <title>Dual space/Orthogonal space/Sum and intersection/Exercise</title>
    <ns>0</ns>
    <id>317276</id>
    <revision>
      <id>2692161</id>
      <timestamp>2024-12-16T11:25:35Z</timestamp>
      <contributor>
        <username>Bocardodarapti</username>
        <id>289675</id>
      </contributor>
      <comment>New resource with &quot;{{ Mathematical text/Exercise |Text= Let {{mat|term= V|pm=}} be a {{ Definitionlink |Premath=K |vector space| |Context=| |pm=, }} and let {{ Relationchain |  U_1,U_2  | \subseteq | V ||  ||  ||  ||  |pm= }} denote {{ Definitionlink |linear subspaces| |Context=| |pm=. }} Show that in the {{ Definitionlink |dual space| |Context=| |pm= }} {{mat|term= {{op:Dual space|V|}} |pm=,}} the equality {{ Relationchain/display |  {{op:Orthogonal space| {{mabr| U_1 + U_2 |}} |}}  ||  {...&quot;</comment>
      <origin>2692161</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="605" sha1="tbmxf7uxbw56gt015gr6ilbp14i4x4b" xml:space="preserve">{{
Mathematical text/Exercise
|Text=
Let {{mat|term= V|pm=}} be a
{{
Definitionlink
|Premath=K
|vector space|
|Context=|
|pm=,
}}
and let
{{
Relationchain
|  U_1,U_2 
| \subseteq | V
|| 
|| 
|| 
|| 
|pm=
}}
denote
{{
Definitionlink
|linear subspaces|
|Context=|
|pm=.
}}
Show that in the
{{
Definitionlink
|dual space|
|Context=|
|pm=
}}
{{mat|term= {{op:Dual space|V|}} |pm=,}} the equality
{{
Relationchain/display
|  {{op:Orthogonal space| {{mabr| U_1 + U_2 |}} |}} 
||  {{op:Orthogonal space|U_1|}}  \cap  {{op:Orthogonal space|U_2|}} 
||
||
||
|pm=
}}
holds.
|Textform=Exercise 
|Category=
|Marks=
}}</text>
      <sha1>tbmxf7uxbw56gt015gr6ilbp14i4x4b</sha1>
    </revision>
  </page>
  <page>
    <title>Complex Analysis/Identity Theorem</title>
    <ns>0</ns>
    <id>317277</id>
    <revision>
      <id>2692163</id>
      <timestamp>2024-12-16T11:27:34Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <comment>New resource with &quot;The Identity Theorem is a statement about holomorphic functions, asserting that they are uniquely determined under relatively weak conditions.  ==Statement== Let &lt;math&gt;U\subseteq \mathbb C&lt;/math&gt; be a domain. For two [[w:en:Holomorphic function|holomorphic]] functions &lt;math&gt;f,g \colon U \to \mathbb C&lt;/math&gt;, the following are equivalent:  (1) &lt;math&gt;f = g&lt;/math&gt; (i.e., &lt;math&gt;f(x) = g(x)&lt;/math&gt; for all &lt;math&gt;x \in U&lt;/math&gt;)  (2) There exists a &lt;math&gt;z_0 \in U&lt;/math&gt; such t...&quot;</comment>
      <origin>2692163</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="6357" sha1="c2ifm8n1uvyhts97uokmekuhsvmxb4u" xml:space="preserve">The Identity Theorem is a statement about holomorphic functions, asserting that they are uniquely determined under relatively weak conditions.

==Statement==
Let &lt;math&gt;U\subseteq \mathbb C&lt;/math&gt; be a domain. For two [[w:en:Holomorphic function|holomorphic]] functions &lt;math&gt;f,g \colon U \to \mathbb C&lt;/math&gt;, the following are equivalent:

(1) &lt;math&gt;f = g&lt;/math&gt; (i.e., &lt;math&gt;f(x) = g(x)&lt;/math&gt; for all &lt;math&gt;x \in U&lt;/math&gt;)

(2) There exists a &lt;math&gt;z_0 \in U&lt;/math&gt; such that &lt;math&gt;f^{(n)}(z_0) = g^{(n)}(z_0)&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;.

(3) The set &lt;math&gt;{z \in U: f(z) = g(z)}&lt;/math&gt; has a limit point in &lt;math&gt;U&lt;/math&gt;.


==Proof==
By considering &lt;math&gt;f-g&lt;/math&gt;, we may assume without loss of generality that &lt;math&gt;g=0&lt;/math&gt;. Equivalently, the proof is reduced to showing the following three statements:

*(N1) &lt;math&gt;f = 0&lt;/math&gt; (i.e., &lt;math&gt;f(x) = 0&lt;/math&gt; for all &lt;math&gt;x \in U&lt;/math&gt;)

*(N2) There exists a &lt;math&gt;z_0 \in U&lt;/math&gt; such that &lt;math&gt;f^{(n)}(z_0) = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;.

*(N3) The zero set &lt;math&gt;N_f := {z \in U: f(z) = 0}&lt;/math&gt; has a limit point in &lt;math&gt;U&lt;/math&gt;.

=== Proof Type ===
The equivalence is proven using a cyclic implication:
&lt;math&gt;(1) \Rightarrow (2) \Rightarrow (3) \Rightarrow (1)&lt;/math&gt;

=== Proof (N1 to N2) ===
(N1) &lt;math&gt;\Rightarrow&lt;/math&gt; (N2) is obvious, as all derivatives of the zero function &lt;math&gt;f&lt;/math&gt; are zero.

=== Proof (N2 to N3) ===
Assume (N2). Consider the power series expansion &lt;math&gt;f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n&lt;/math&gt; in &lt;math&gt;B_r(z_0)&lt;/math&gt; with &lt;math&gt;r &gt; 0&lt;/math&gt;.
Here, &lt;math&gt;a_n = \frac{f^{(n)}(z_0)}{n!} = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;.
Thus, &lt;math&gt;f|_{B_r(z_0)} = 0&lt;/math&gt;, and (N3) follows.

=== Proof (N3 to N1) – Contradiction Proof ===
The step (N3) &lt;math&gt;\Rightarrow&lt;/math&gt; (N1) is proven by contradiction. Assume the zero set has a limit point and &lt;math&gt;f&lt;/math&gt; is not the zero function.

=== Proof 1 - (N3 to N1) - Power Series Expansion at Limit Point ===
Assume (N3), i.e., the set &lt;math&gt;N_f&lt;/math&gt; of zeros of &lt;math&gt;f&lt;/math&gt; has a limit point &lt;math&gt;z_0 \in U&lt;/math&gt;.
Thus, there exists a sequence &lt;math&gt;(z_n) \in U^{\mathbb N}&lt;/math&gt; with &lt;math&gt;z_n \to z_0&lt;/math&gt; and &lt;math&gt;f(z_n) = 0&lt;/math&gt; as well as &lt;math&gt;z_n \ne z_0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;. Let &lt;math&gt;f(z) = \sum_{n=0}^\infty a_n (z-z_0)^n&lt;/math&gt; be the power series expansion of &lt;math&gt;f&lt;/math&gt; around &lt;math&gt;z_0&lt;/math&gt;.

=== Proof 2 - (N3 to N1) - Power Series Expansion ===
Suppose there exists &lt;math&gt;n \in \mathbb N&lt;/math&gt; with &lt;math&gt;a_n \ne 0&lt;/math&gt;. Due to the well-ordering property of &lt;math&gt;\mathbb{N}&lt;/math&gt;, there would also be a smallest such &lt;math&gt;n&lt;/math&gt;. Then

&lt;center&gt;&lt;math&gt; f(z) = (z-z_0)^n \sum_{k=0}^\infty a_{n+k}(z-z_0)^k, \qquad |z-z_0| &lt; r, a_n \ne 0 &lt;/math&gt;&lt;/center&gt;  
=== Proof 3 - (N3 to N1) - Power Series Evaluation ===
For each &lt;math&gt;i \in \mathbb N&lt;/math&gt;, we have

&lt;center&gt;&lt;math&gt; 0 = f(z_i) = (z_i-z_0)^n \sum_{k=0}^\infty a_{n+k}(z_i - z_0)^k &lt;/math&gt;&lt;/center&gt;  
=== Proof 4 - (N3 to N1) - Limit Process ===
Since &lt;math&gt;z_i \ne z_0&lt;/math&gt; and &lt;math&gt;\displaystyle \lim_{i\to \infty} z_i = z_0&lt;/math&gt;, we get

&lt;center&gt;&lt;math&gt; 0 = \sum_{k=0}^\infty a_{n+k} (z_i - z_0)^k \to a_n, \qquad i \to \infty &lt;/math&gt;&lt;/center&gt;  
As &lt;math&gt;a_{n+k} (z_i - z_0)^k \to 0&lt;/math&gt; for all &lt;math&gt;k &gt; 0&lt;/math&gt; as &lt;math&gt;i \to \infty&lt;/math&gt;. This contradicts &lt;math&gt;a_n \ne 0&lt;/math&gt;. Therefore, &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and hence &lt;math&gt;f^{(n)}(z_0) = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb N&lt;/math&gt;, i.e., (N2) holds. 
=== Proof 5 - (N3 to N1) - V is Closed ===
If (N2) holds, set &lt;math&gt;V := \bigcap_{n\ge 0} { z \in U , | , f^{(n)}(z)=0}&lt;/math&gt;.
&lt;math&gt;V&lt;/math&gt; is closed in &lt;math&gt;U&lt;/math&gt; as the intersection of closed sets, because the &lt;math&gt;f^{(n)}&lt;/math&gt; are continuous, and preimages of closed sets (here &lt;math&gt;{0}&lt;/math&gt;) are closed.

=== Proof 6 - (N3 to N1) - V is Open ===
&lt;math&gt;V&lt;/math&gt; is also open in &lt;math&gt;U&lt;/math&gt;, as for every &lt;math&gt;w \in V&lt;/math&gt;, the power series expansion of &lt;math&gt;f&lt;/math&gt; around &lt;math&gt;w&lt;/math&gt; vanishes. Thus, &lt;math&gt;f&lt;/math&gt; is locally zero around &lt;math&gt;w&lt;/math&gt;. Since &lt;math&gt;z_0 \in V&lt;/math&gt;, &lt;math&gt;V&lt;/math&gt; is non-empty, and hence &lt;math&gt;V = U&lt;/math&gt; due to the connectedness of &lt;math&gt;U&lt;/math&gt;.

=== Proof 7 - From (N1)-(N3) to (1)-(3) ===
The statement of the Identity Theorem (1)-(3) follows for arbitrary &lt;math&gt;g:U \to \mathbb{C}&lt;/math&gt; and &lt;math&gt;h:U \to \mathbb{C}&lt;/math&gt;, by applying (N1)-(N3) to &lt;math&gt;f:=g-h&lt;/math&gt;.

==See Also==

*[[w:en:Complex Analysis|Complex Analysis]]
*[[w:en:Casorati-Weierstrass theorem|Casorati-Weierstrass Theorem]]
== Page Information ==
This learning resource can be presented as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Course:%20Complex%20Analysis/Identity%20Theorem&amp;author=Course:%20Complex%20Analysis&amp;language=en&amp;audioslide=yes Wiki2Reveal Slide Set]'''.

=== Wiki2Reveal ===

This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Course:%20Complex%20Analysis/Identity%20Theorem&amp;author=Course:%20Complex%20Analysis&amp;language=en&amp;audioslide=yes Wiki2Reveal Slide Set]''' was created for the learning unit '''[https://en.wikiversity.org/wiki/_Course:%20Complex%20Analysis Course: Complex Analysis]'''' using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator].
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&lt;!-- * Next content of the course is [[]] --&gt;

[[Category:Wiki2Reveal]]
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&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;11&lt;/span&gt;&lt;/span&gt;
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&lt;span class=&quot;number-of-blocks&quot;&gt;Number of blocks: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;44&lt;/span&gt;&lt;/span&gt;
Integer partition: &amp;nbsp; &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;1&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;3&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;4&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;8&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;6&lt;/span&gt; + &lt;span class=&quot;count&quot;&gt;12&lt;/span&gt;⋅&lt;span class=&quot;size&quot;&gt;12&lt;/span&gt;
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[[Category:Boolf prop/3-ary|dominion]]</text>
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  <page>
    <title>Complex Analysis/Liouville's Theorem</title>
    <ns>0</ns>
    <id>317281</id>
    <revision>
      <id>2692172</id>
      <timestamp>2024-12-16T11:56:44Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <comment>New resource with &quot;The Liouville Theorem is a statement about [[w:en:Holomorphic function|holomorphic]] functions defined on the entire complex plane &lt;math&gt;\mathbb C&lt;/math&gt;.  ==Statement== Let &lt;math&gt;f \colon \mathbb C \to \mathbb C&lt;/math&gt; be holomorphic and bounded. Then &lt;math&gt;f&lt;/math&gt; is constant.  ==Proof== For every &lt;math&gt;R &gt; 0&lt;/math&gt; and every &lt;math&gt;z \in \mathbb C&lt;/math&gt;, we have by the [[w:en:Cauchy's integral formula#Corollaries|Cauchy integral formula]]:  &lt;center&gt;&lt;math&gt;   \begin{ar...&quot;</comment>
      <origin>2692172</origin>
      <model>wikitext</model>
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      <text bytes="2865" sha1="15e6rpkompxzn9ljnywuq6zzb757xd2" xml:space="preserve">The Liouville Theorem is a statement about [[w:en:Holomorphic function|holomorphic]] functions defined on the entire complex plane &lt;math&gt;\mathbb C&lt;/math&gt;.

==Statement==
Let &lt;math&gt;f \colon \mathbb C \to \mathbb C&lt;/math&gt; be holomorphic and bounded. Then &lt;math&gt;f&lt;/math&gt; is constant.

==Proof==
For every &lt;math&gt;R &gt; 0&lt;/math&gt; and every &lt;math&gt;z \in \mathbb C&lt;/math&gt;, we have by the [[w:en:Cauchy's integral formula#Corollaries|Cauchy integral formula]]:

&lt;center&gt;&lt;math&gt;
  \begin{array}{rl}
    |f'(z)| &amp;= \displaystyle \frac 1{2\pi} \left|\int_{\partial B_R(z)} \frac{f(w)}{(w-z)^2}\, dw\right|\\
            &amp;\le \displaystyle \frac 1{2\pi} 2\pi R \cdot \frac 1{R^2} \cdot  \sup_{w \in B_R(z)} |f(w)|\\
            &amp;\le \frac M{R}\\
            &amp;\to 0, \qquad R \to \infty
  \end{array}
&lt;/math&gt;&lt;/center&gt;
Thus, &lt;math&gt;f' = 0&lt;/math&gt;, and therefore &lt;math&gt;f&lt;/math&gt; is constant.  

==See Also==

*[[w:en:Complex Analysis|Complex Analysis]]

*[[w:en:Cauchy's integral formula#Collaries|Cauchy's integral formula]]
*[[Wiki2Reveal|Wiki2Reveal]]
== Page Information ==
This learning resource can be presented as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Satz%20von%20Liouville&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes&amp;shorttitle=Satz%20von%20Liouville&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''.
=== Wiki2Reveal ===
This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Satz%20von%20Liouville&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes&amp;shorttitle=Satz%20von%20Liouville&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie Kurs:Funktionentheorie]''' using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator].
&lt;!--
* The content of the page is based on the following contents:
** [https://de.wikipedia.org/wiki/Kurs:Funktionentheorie/Satz%20von%20Liouville https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz%20von%20Liouville]
--&gt;
* [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz%20von%20Liouville The page] was created as a document type [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron SLIDE].
* Link to the source in Wikiversity: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Satz%20von%20Liouville
* See also further information about [[w:de:Wiki2Reveal|Wiki2Reveal]] and under [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Satz%20von%20Liouville&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes&amp;shorttitle=Satz%20von%20Liouville&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Link Generator].
&lt;!-- * Next content of the course is [[]] --&gt;;

[[Category:Wiki2Reveal]]</text>
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