Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.44.0-wmf.6 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk 4 28 2690874 2690691 2024-12-08T15:44:49Z Ottawahitech 2369270 /* An unexplained spurt of Wikiversity page views */ Reply 2690874 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&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,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> 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) == 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 "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "open", 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 "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[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, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "container" 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! == <section begin="Sub-referencing"/> [[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) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 == <section begin="announcement-content" /> :''[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&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,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 "you can switch" 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! == <section begin="announcement-content" /> 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<section end="announcement-content" /> [[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) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' 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 "RICH-2K" 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "a test-environment". 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." 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&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "on the map" (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 == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&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.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{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 <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> 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]].<br><br>We welcome your feedback and questions.<br> [[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) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&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 "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&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 "Wikipedia"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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == '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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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: "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." --[[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 "Information sciences" (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 <code>|filing=deep</code> 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]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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: <blockquote>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.</blockquote> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikiversal" 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 "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: 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 == <section begin="announcement-content" /> 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 <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> 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, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&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&diff=prev&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) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> == 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,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" 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) mnm84733vcx3h1x5yn775ootfyi417i 2690899 2690874 2024-12-08T21:05:26Z RockTransport 2992610 /* Wikiversity - Newsletters */ new section 2690899 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&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,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> 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) == 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 "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "open", 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 "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[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, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "container" 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! == <section begin="Sub-referencing"/> [[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) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 == <section begin="announcement-content" /> :''[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&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,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 "you can switch" 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! == <section begin="announcement-content" /> 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<section end="announcement-content" /> [[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) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' 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 "RICH-2K" 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "a test-environment". 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." 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&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "on the map" (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 == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&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.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{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 <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> 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]].<br><br>We welcome your feedback and questions.<br> [[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) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&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 "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&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 "Wikipedia"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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == '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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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: "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." --[[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 "Information sciences" (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 <code>|filing=deep</code> 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]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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: <blockquote>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.</blockquote> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikiversal" 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 "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: 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 == <section begin="announcement-content" /> 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 <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> 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, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&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&diff=prev&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) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> == 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,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" 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) == Wikiversity - Newsletters == Hello All, I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events; which would bolster engagement by many people. I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC) 7vzczyapznrpe3l02xpw883i05lbuzq 2690900 2690899 2024-12-08T21:06:19Z RockTransport 2992610 /* Wikiversity - Newsletters */ 2690900 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&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,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> 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) == 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 "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "open", 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 "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[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, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "container" 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! == <section begin="Sub-referencing"/> [[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) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 == <section begin="announcement-content" /> :''[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&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,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 "you can switch" 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! == <section begin="announcement-content" /> 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<section end="announcement-content" /> [[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) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' 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 "RICH-2K" 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "a test-environment". 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." 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&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "on the map" (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 == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&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.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{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 <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> 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]].<br><br>We welcome your feedback and questions.<br> [[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) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&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 "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&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 "Wikipedia"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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == '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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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: "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." --[[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 "Information sciences" (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 <code>|filing=deep</code> 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]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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: <blockquote>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.</blockquote> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikiversal" 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 "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: 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 == <section begin="announcement-content" /> 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 <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> 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, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&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&diff=prev&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) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> == 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,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" 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) == 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) 9vokkfv0ld3ibgi3zab9611ksu1xhn0 2690901 2690900 2024-12-08T21:06:52Z RockTransport 2992610 /* User group for Wikiversians */ Reply 2690901 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == Reminder! Vote closing soon to fill vacancies of the first U4C == <section begin="announcement-content" /> :''[[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&language=&action=page&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,<section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 <code>{{[[:w:Template:Wikiversity|Wikiversity]]}}</code> 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 "representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans." 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "open", 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 "fit" for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of "learning resources". [[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, "MediaWiki" is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and "Wikimedia Foundation" is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "container" 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! == <section begin="Sub-referencing"/> [[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) <section end="Sub-referencing"/> <!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&oldid=27309345 --> == 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 == <section begin="announcement-content" /> :''[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&group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&language=&action=page&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,<section end="announcement-content" /> [[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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 "you can switch" 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! == <section begin="announcement-content" /> 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<section end="announcement-content" /> [[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) <!-- Message sent by User:RamzyM (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Separate page for hyperbola. == Good morning, I notice that a search for "hyperbola" redirects to "Conic sections". At present there is a separate page for "ellipse". Therefore a separate page for "hyperbola" seems to be justified. Could this redirection be changed so that search for "hyperbola" goes to a separate page for "hyperbola"? 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 ''<nowiki>https://en.wikiversity.org/wiki/RICH-2K</nowiki>'' 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 "RICH-2K" 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "a test-environment". 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "The" is acceptable, but if you want it to sort alphabetically, you will have to use <nowiki>{{DEFAULTSORT:}}</nowiki>. E.g. to get Category:The Best Stuff to sort under "B", insert "<nowiki>{{DEFAULTSORT:Best Stuff, The}}</nowiki>. :*Trailing "etc." 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&nbsp;years. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "on the map" (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 == <section begin="server-switch"/><div class="plainlinks"> [[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&group=page-Tech%2FServer+switch&language=&action=page&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.'''</div><section end="server-switch"/> [[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC) <!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27248326 --> == 'Wikidata item' link is moving. Find out where... == <div lang="en" dir="ltr" class="mw-content-ltr"><i>Apologies for cross-posting in English. Please consider translating this message.</i>{{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 <span style="color: #54595d;"><u>''General''</u></span> section of the '''Tools''' sidebar menu will move into the <span style="color: #54595d;"><u>''In Other Projects''</u></span> 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]].<br><br>We welcome your feedback and questions.<br> [[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) </div> <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27524260 --> ==Download as PDF== [[Phabricator:T376438]]: "Download to PDF" on en.wv is returning error: "{"name":"HTTPError","message":"500","status":500,"detail":"Internal Server Error"}" -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 22:36, 3 October 2024 (UTC) :I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikipedia" is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&limit=500&offset=0&ns828=1&search=Wikipedia&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 "Wikiversity" ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&diff=2662815&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 "Wikipedia"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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27510935 --> == '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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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: "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." --[[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 "Information sciences" (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 <code>|filing=deep</code> 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]], "Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely"; 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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: <blockquote>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.</blockquote> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 "Wikiversal" 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 "the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit." ::: 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 == <section begin="announcement-content" /> 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 <section end="announcement-content" /> [[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC) <!-- Message sent by User:MPossoupe (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27183190 --> == Seeking volunteers to join several of the movement’s committees == <section begin="announcement-content" /> 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, <section end="announcement-content" /> -- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27601062 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 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&source=pageviews&agent=user&start=2024-06-01&end=2024-10-18&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&diff=prev&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) <!-- Message sent by User:Danny Benjafield (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Danny_Benjafield_(WMDE)/MassMessage_Test_List&oldid=27535421 --> == 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,<br> The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC) <!-- Message sent by User:ZI Jony@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Non-Technical_Village_Pumps_distribution_list&oldid=27568454 --> == 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]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 00:56, 31 October 2024 (UTC) à == Import Resource From Wikibooks? == Hello! [[wikibooks:Character_List_for_Baxter&Sagart|Character List for Baxter&Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 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 <https://zonestamp.toolforge.org/1732896000>). If you're interested in joining, you can sign up on this wiki page: <https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024>. 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) <!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=27746256 --> == Events on Wikiversity == Since Wikipedia and Wikivoyage are having their "Asian Month" editathon, I was thinking if we could start up a Wikiversity version of that. This would be an "Asian Month" 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) == 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) 2la0roq8kpvlfin6pfanqyu719bua6j 0 2103 2690898 2339580 2024-12-08T20:56:33Z RockTransport 2992610 Minor change 2690898 wikitext text/x-wiki [[File:Hello World Brian Kernighan 1978.jpg|thumb|right|Hello World! by Brian Kernighan. Based on a 1978 Bell Laboratories internal memorandum by Brian Kernighan, Programming in C: A Tutorial, which contains the first known version.]] As described in more detail in [[w:"Hello,_World!"_program|the related Wikipedia article]], '''Hello, world!''' is a classic "first program" one creates when learning a new programming language. The objective of the application is the same: to print the text "Hello, world!" to the screen in some form, be it console output or a dialog. In many cases, the statement required to do this is just a single line. It seems appropriate that our introduction to Computer Science occupied this title. As a student, the first choice to make is to decide ''what kind of knowledge you are looking for''. Of course, this depends upon your needs. You might be: * A learned computer scientist or professional eager to contribute research and course material * Computer professional seeking an alternative to expensive commercial certification * Adult non-computer professional or entrepreneur who could benefit from academic/practical knowledge of computing * College-eligible (or not) student considering a degree * Casual user trying to to catch/spread the next [[w:Computer virus|virus]] * Hobbyist or computer gamer looking to get the most out of your computing experience * Complete newbie looking for a place to start This is an exciting time for education, and for those of us wishing to collaborate and share knowledge, skills and experience. At present, we are only limited by the sky, and some very large hard drives in a server farm somewhere. {{TOC right}} == Examples of ''Hello, world!'' == === [[Hello World/Ada|Ada]] === <syntaxhighlight lang="ada"> with Ada.Text_IO; procedure Hello is begin Ada.Text_IO.Put_Line ("Hello, world!"); end Hello; </syntaxhighlight> For an explanation see [[b:Ada Programming:Basic]]. === ASP === <syntaxhighlight lang="asp"> <% Response.Write "Hello, world!" %> </syntaxhighlight> or <syntaxhighlight lang="asp"> <%="Hello, World!"%> </syntaxhighlight> === Alef++ === <syntaxhighlight lang="text"> sub say : void { System->out->println[ $0#0 ]; } main{ say[Hello, world!]; } </syntaxhighlight> === [[Topic:Assembly language|Assembly]] === ''x86 compatible'' for [[Wikipedia:MS-DOS|MS-DOS]]. <syntaxhighlight lang="asm"> title Hello World Program dosseg .model small .stack 100h .data hello_message db 'Hello, world!',0dh,0ah,'$' .code main proc mov ax,@data mov ds,ax mov ah,9 mov dx,offset hello_message int 21h mov ax,4C00h int 21h main endp end main </syntaxhighlight> === BASH === <syntaxhighlight lang="bash"> #!/bin/bash echo "Hello, world!" </syntaxhighlight> === BASIC === ==== Applesoft BASIC ==== ''Used on Apple ][ machines (Apple ][+, ][e, //c, ][GS)'' <syntaxhighlight lang="qbasic"> 10 PRINT "HELLO, WORLD!" </syntaxhighlight> -or- <syntaxhighlight lang="qbasic"> 10 ? "HELLO, WORLD!" </syntaxhighlight> ==== Bally/Astrocade Basic ==== ''As used on the Bally and Astrocade game systems ca. 1978'' <syntaxhighlight lang="qbasic"> 10 PRINT "HELLO, WORLD!" </syntaxhighlight> ==== Commodore BASIC ==== ''As used on a Commodore 64, ca. 1984'' <syntaxhighlight lang="qbasic"> 10 ? "Hello, world!" </syntaxhighlight> ==== Dark Basic ==== <syntaxhighlight lang="qbasic"> PRINT "Hello, world!" </syntaxhighlight> ==== FreeBASIC and QuickBASIC ==== <syntaxhighlight lang="qbasic"> PRINT "Hello, world!" SLEEP </syntaxhighlight> or: <syntaxhighlight lang="qbasic"> ? "Hello, world!" sleep </syntaxhighlight> ==== Intellivision Basic ==== ''As used on a Mattel Intellivision, ca. 1983'' <syntaxhighlight lang="qbasic"> 10 PRINT "HELLO, WORLD!" </syntaxhighlight> ==== Intellivision ECS Basic ==== ''As used in the Mattel Intellivision ECS'' <syntaxhighlight lang="qbasic"> 10 PRIN "HELLO, WORLD." </syntaxhighlight> ''! not on ECS keyboard. Only 4 char. commands in ECS Basic'' ==== Liberty BASIC ==== <syntaxhighlight lang="qbasic"> print "Hello, world!" </syntaxhighlight> === Batch === <syntaxhighlight lang="bash"> echo Hello, world! </syntaxhighlight> === [[C]] === <syntaxhighlight lang="c"> #include <stdio.h> int main(void) { printf( "Hello, world!\n" ); return 0; } </syntaxhighlight> === [[Topic:C Sharp|C#]] === <syntaxhighlight lang="csharp"> using System; namespace HelloWorld { class Program { static void Main() { Console.WriteLine("Hello, world!"); } } } </syntaxhighlight> === [[C++]] === <syntaxhighlight lang="cpp"> #include <iostream> using namespace std; int main() { cout << "Hello, world!\n"; return 0; } </syntaxhighlight> === COBOL === <syntaxhighlight lang="cobol"> IDENTIFICATION DIVISION. PROGRAM-ID. HELLO-WORLD. PROCEDURE DIVISION. DISPLAY 'Hello, world'. STOP RUN. </syntaxhighlight> === Common Lisp === <syntaxhighlight lang="lisp"> (print "Hello, world!") </syntaxhighlight> Or: <syntaxhighlight lang="lisp"> (format t "Hello, world!~%") </syntaxhighlight> === [[Delphi]] === <syntaxhighlight lang="delphi"> begin Writeln('Hello, world!'); end. </syntaxhighlight> === Eztrieve (IBM Mainframe programming language). === <code><pre>JOB NULL DISPLAY "HELLO, WORLD" STOP</pre></code> === [[Forth]] === <code><pre>: HELLO ." Hello, world!" ; HELLO</pre></code> === [[Fortran]] === <syntaxhighlight lang="fortran"> PROGRAM HELLO PRINT *,'Hello, world' STOP END </syntaxhighlight> === [[Go]] === <syntaxhighlight lang="go"> package main import "fmt" func main() { fmt.Println("Hello, World") } </syntaxhighlight> === Haskell === <code><pre>main :: IO () main = putStrLn "Hello, world!"</pre></code> === [[HTML|Html]] === <syntaxhighlight lang="html4strict"> <html> <head> <title>Hello, world!</title> </head> <body> <p> Hello, world! </p> </body> </html> </syntaxhighlight> === [[Java]] === <syntaxhighlight lang="java"> class HelloWorldApp { public static void main(String[] args) { System.out.println("Hello World!"); // Display the string. } } </syntaxhighlight> === [[Portal:JavaScript|JavaScript]] (aka JScript, ECMAScript, LiveScript) === <syntaxhighlight lang="java"> document.println("Hello, world!"); </syntaxhighlight> or <syntaxhighlight lang="java"> alert("Hello, world!"); </syntaxhighlight> or <syntaxhighlight lang="java"> document.writeln("Hello, world!"); </syntaxhighlight> === [[Luka]] === <pre> print "Hello, world" </pre> or, with proper syntax <pre> print( "Hello, world!" ); </pre> === [[w:Oberon (programming language)|Oberon]] === <pre>MODULE Hello; IMPORT Out; PROCEDURE World*; BEGIN Out.Open; Out.String("Hello, world!"); Out.Ln; END World; END Hello.</pre> === [[OCaml]] === <syntaxhighlight lang="ocaml"> print_endline "Hello, world!" </syntaxhighlight> === [[Pascal]] === <syntaxhighlight lang="pascal"> program HelloWorld; begin writeln( 'Hello, world!' ); end. </syntaxhighlight> === [[Portal:Perl|Perl]] === <syntaxhighlight lang="perl"> #!/usr/bin/perl print "Hello, world!\n"; </syntaxhighlight> === [[Portal:PHP|PHP]] === <syntaxhighlight lang="php"> <?php echo "Hello, world!"; ?> </syntaxhighlight> or (with short_tags enabled in php.ini) <syntaxhighlight lang="php"> <? echo "Hello, world!"; ?> </syntaxhighlight> or (with asp_tags enabled in php.ini) <syntaxhighlight lang="php"> <% echo "Hello, world!"; %> </syntaxhighlight> or <syntaxhighlight lang="php"> <?="Hello, world!"?> </syntaxhighlight> === [[Topic:Python|Python]] === With Python 2 <syntaxhighlight lang="python"> #!/usr/bin/env python print 'Hello, world!' </syntaxhighlight> Or with Python 3 <syntaxhighlight lang="python"> print("Hello, world!") </syntaxhighlight> The first line is used on Unix systems only, and is optional even there. The advantage is that it allows the file to be invoked directly (if <code>chmod +x</code>), without explicitly specifying the <code>python</code> interpreter. === [[Ruby]] === <syntaxhighlight lang="ruby"> puts 'Hello, world!' </syntaxhighlight> Another way to do it, albeit more obscure: <syntaxhighlight lang="ruby"> #!/usr/local/bin/ruby puts 1767707668033969.to_s(36) </syntaxhighlight> === [[Tcl]] === <syntaxhighlight lang="tcl"> #!/usr/bin/tclsh puts "Hello, world!" </syntaxhighlight> === [[Trekkie]] === <pre> "Computer?" *Bee bee boo "Create program 'Hello, World! Picard-alpha-1'" *Boo boo bee "Parameters: Display the phrase 'Hello, world!' on the screen the program is executed from until the program is terminated." *Bee bee "Save program." *Boo bee boo </pre> === [[Turing]] === <pre> put "Hello World!" </pre> === [[Visual Basic|Visual Basic 6]] === <syntaxhighlight lang="vb"> Sub Form1_Load() MsgBox "Hello, world!" End Sub </syntaxhighlight> == Assignment == Create a '''Hello, world!''' program in a language not listed above, then edit this page and add it to the collection. === Visual Basic .NET === <syntaxhighlight lang="vbnet"> Module Module1 Sub Main() Console.WriteLine("Hello, world!") End Sub End Module </syntaxhighlight> === C === Because the tradition of using the phrase "[[w:"Hello,_World!"_program|Hello, world]]!" as a test message was influenced by an example program in the seminal book ''[[w:The C Programming Language (book)|The C Programming Language]]''.<ref>{{cite book | last = Kernighan | first = Brian W. | authorlink = w:Brian W. Kernighan |author2=w:Ritchie, Dennis M. | title = The C Programming Language | edition = 1st | publisher = [[Prentice Hall]] | date = 1978 | location = [[Englewood Cliffs, NJ]] | isbn = 0-13-110163-3 | authorlink2 = Dennis M. Ritchie }}</ref> that original example is reproduced here. <syntaxhighlight lang="text"> #include <stdio.h> main( ) { printf("hello, world\n"); } </syntaxhighlight> === '''LOLCODE''' === <syntaxhighlight lang="text"> HAI CAN HAS STDIO? VISIBLE "Hello world!" KTHXBYE </syntaxhighlight> === '''Natural ''' === <pre> WRITE 'Hello, world!' END </pre> <big>Hello, world!</big> === '''XML''' === <syntaxhighlight lang="xml"> <?xml version="1.0"?> <hello> <messagename="Hello" /> <message> Hello, World! </message> </hello> </syntaxhighlight> Or with attributes: <syntaxhighlight lang="xml"> <?xml version="1.0"?> <hello messagename="Hello, World!"> Hello, world! </hello> </syntaxhighlight> === '''[[w:MACRO-11|MACRO-11]]''' === <syntaxhighlight lang="text"> .TITLE HELLO WORLD .MCALL .TTYOUT,.EXIT HELLO:: MOV #MSG,R1 ;STARTING ADDRESS OF STRING 1$: MOVB (R1)+,R0 ;FETCH NEXT CHARACTER BEQ DONE ;IF ZERO, EXIT LOOP .TTYOUT ;OTHERWISE PRINT IT BR 1$ ;REPEAT LOOP DONE: .EXIT MSG: .ASCIZ /Hello, world!/ .END HELLO </syntaxhighlight> == More about Computer Programming == *[[Portal:Computer programming|Topic:Computer Programming]] ==See also== {{wikipedia2|Hello world program}}{{commonscat|Hello World}} * [https://web.archive.org/web/20150404011657/http://en.wikipedia.org/wiki/Hello_world_program_examples Hello world program examples] from Wikipedia (archived copy) ==External links== * [http://helloworldcollection.de The Hello World Collection] with 500+ Hello World programs [[Category:Computer programming]] [[Category:Programming languages]] [[cs:Hello world!]] obnd2l0jwe19gnou09z6k0ugcwgsz57 0 53432 2690909 2690482 2024-12-08T21:44:54Z Atcovi 276019 fix 2690909 wikitext text/x-wiki '''2024''' * '''December 5: [[Pedophilia: Innate or Learned?|''Pedophilia: Innate or Learned?'']]''' has been published and is ready for viewers. *'''July 22:''' [[Wikiphilosophers]] has launched on Wikiversity. The goal of this project is to provide an overview of all philosophical ideas of philosophers and thinkers within Wikiversity. *'''May 13:''' [[Motivation and emotion/Book/About/Collaborative authoring using wiki|Collaborative authoring using wiki: An open education case study]] published in the ''International Journal for Students as Partners'' *'''March 15:''' [[User:Jtneill/Presentations/Using open wikis for teaching and learning|Using open wikis for teaching and learning]] presentation to the ASCILITE Learning Design Special Interest Group *'''January 5:''' The [[Real Good Religion]] course is now welcoming students seeking adventure <noinclude> [[Category:Main page templates]] [[Category:News]] [[Category:Wikiversity news]] </noinclude> k88xh7w34ycujj8f864lejyca6ih7ku 102 57504 2690989 853268 2024-12-09T08:43:38Z RockTransport 2992610 punctuation 2690989 wikitext text/x-wiki {{image|name=AF-kindergarten.jpg|width=250px|pad=10px|caption= |float=left}} '''Welcome''' to Wikiversity's Pre-School Education Portal. This page is a place to go for preschool and early learning content. '''[[Pre-school education]]''' is usually for children between the ages of about two or three up to five or six years. [[Portal:Primary Education|Primary education]] comes after Pre-School education. Toddlers, young children and their parents and teachers are welcome to create and share learning materials using this [[wikimedia:|Wikimedia Foundation]] website for global learning. lfmdkbp8xmz5kvbq5vm2we7l7cgdw2v 1 67093 2690902 2596259 2024-12-08T21:22:36Z Watchduck 137431 2690902 wikitext text/x-wiki === Semi === Should this page be full protected? Obviously all the custodians should have it watchlisted, but no one other than a custodian has ever edited it. [[User:Salmon of Doubt|Salmon of Doubt]] 11:44, 13 September 2008 (UTC) * The idea of the complex protection system on the various main page templates was to leave this particular page semi-protected only so that ordinary users can add news items. This page should only be fully protected if it becomes a repeated target for vandalism, which is not yet the case. --[[User:McCormack|McCormack]] 12:07, 13 September 2008 (UTC) == Possible use of dynamicpagelist == {{#invoke:DynamicPageList |show |category=Completed resources |namespace= |count=10 |mode=unordered |ordermethod=categoryaddlastedit |orderdate=categoryaddlastdate |order=descending }} --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:19, 16 October 2016 (UTC) So, if I follow current custom, [[Main Page/News]] would appear something like: '''2016''' * '''15 October:''' The new courses [[IT Fundamentals]] and [[Assessing Human Rights]] are completed and ready for students and professionals * '''15 October:''' The new book chapter [[Motivation and emotion/Book/2016/Emotional hijacking|Emotional hijacking]] is ready for use by psychologists and their students * '''12 October:''' The Python Programming lessons [[Python Programming/Modules|Modules]] and [[Python Programming/Classes|Classes]] are completed and ready for students * '''15 August:''' '''Wikiversity celebrates its 10-year anniversary!''' * '''15 August:''' The fall semester begins for the course [[principles of radiation astronomy]]. * '''8 August:''' The [[ENG 099|open course in conversational American English]] for EFL/ESL/ELL/ESOL students starts. * '''31 July:''' For [[Wikiversity:Year of Science 2016|'''The Wikimedia Year of Science''']], try using the May 9, 2016 [[w:Transit of Mercury]] to improve the [[Stars/Sun/Locating the Sun|location of the Sun]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:22, 16 October 2016 (UTC) ::I'm confused. You wrote that "It would be great to have five to six things to put in the news each week instead of five to six for three to four months!" Are you also wanting to maintain this manually rather than using DynamicPageList to do the updates automatically? If you are working manually, there is no reason to list IT Fundamentals again, as it is only on the DynamicPageList due to vandalism. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:40, 16 October 2016 (UTC) :::Originally I was intending to try to maintain this subpage manually. Your use of dynamicpagelist suggests that at least this part could be automated. But there appears to be no easy way to include what the resources are, their approximate dates of completion and intended targets. I also wanted to include those News items that are not in that specific category. Combining the two in an automated way would really be great! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:11, 17 October 2016 (UTC) ==Earlier unannounced completed resources== These would currently have been updated or improved: {{div col|colwidth=12em}} {{#invoke:DynamicPageList |show |category=Completed resources |namespace= |count=10 |mode=unordered |ordermethod=categoryaddlastedit |orderdate=categoryaddlastdate |order=descending }} {{Div col end}} --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:54, 25 June 2017 (UTC) # [[Mathematical Properties]] was announced on 20 January 2018, # [[Internet Protocol Analysis]] was announced on 15 March 2019, # [[Overcoming Hate]] was announced on 9 July 2019, # [[Unleashing Creativity]] was announced on 2 July 2019, # [[Candor]] was announced on 28 June 2019, # [[Exploring Social Constructs]] was announced on 31 May 2019, # [[Applied Programming]] was announced on 17 May 2018, # [[Finding Courage]] was announced on 17 April 2019, # [[Moral Reasoning]] was announced on 14 April 2019, # [[Coping with Ego]] was announced on 17 March 2019, # [[Resolving Anger]] was announced on 1 February 2019, # [[Communicating Power]] was announced on 29 January 2019, # [[What you can change and what you cannot]] was announced on 28 January 2019, # [[Recognizing Emotions]] was announced on 25 January 2019, # [[Appraising Emotional Responses]] was announced on 24 January 2019, and # [[Open Source 3-D Printing]] was announced on 20 January 2019. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 01:35, 15 September 2019 (UTC) == COVID-19 COBOL programmer shortage? == <blockquote>COVID-19 COBOL programmer shortage</blockquote> Where can I find information about this topic (from 12 April news item)? Thanks in advance. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 19:13, 21 April 2020 (UTC) :{{Ping|Ottawahitech}} There's quite a bit you can learn with a search engine. Just from my cursory following of the news, I know that the problem is particularly acute in New Jersey in the banking industry. If you're interested in learning, you may want to track down *How to Learn COBOL in 21 Days*, which is free online. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:28, 21 April 2020 (UTC) :{{At|Ottawahitech}} A long list of additional resources are included at [[Talk:COBOL]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:29, 21 April 2020 (UTC) ::{{At|Koavf|Dave Braunschweig}} actually there is a story behind my question. Back in 2011 I was trying to insert information into a Bank of America article and was meeting a [https://en.wikipedia.org/w/index.php?oldid=436375222&title=Talk%3ABank_of_America#The_suicide_of_Kevin_Flanagan lot of resistance], even though at the time the information was documented on another Wikipedia article where I first learned about it. The information was about a computer programmer who committed suicide in front of one of Bank of America's properties. According to the the article, he was despondent after losing his job at Bofa, which went to H1B workers. Flanagn and his layed off coleauges all of whom lost their jobs were tasked with training their replacements. ::To make a long story short, there were two persistent editors guarding information on the Bofa article, and I eventually moved on to other areas. Those two kept the story out of wikipedia, and months or even years later I found out that the article about Flanagan himself disappreaed as well. Since then I am skeptical about claims of a shortage of skilled people made by employers without the corresponding verifiable reliable sources showing a shortage actually exists. Maybe that's just me? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 02:24, 22 April 2020 (UTC) :::{{ping|Ottawahitech}} Well, that certainly escalated quickly. Without getting into the specifics of any editing disputes from the past or possible criminal disappearances, etc. it seems like your main question is, "Are there ''really'' a shortage of COBOL specialists?" and the answer is 100% yes. This has been a known problem for well over a decade (likely longer) and if you want evidence of that, I would recommend looking at job listings to see how much a company will pay someone with COBOL knowledge versus (e.g.) Python or C+. Having proficiency and especially experience in COBOL is a good way to make money in programming. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:37, 22 April 2020 (UTC) :::{{At|Ottawahitech}} I'm sorry, but for me this approach crosses the line. I have recommended that you move on from past incidents on other Wikimedia projects.[https://en.wikiversity.org/w/index.php?title=User_talk%3AOttawahitech&type=revision&diff=2141168&oldid=2137899] I have strongly encouraged you to move on from past incidents on other Wikimedia projects. I am now warning you to move on from past incidents on other Wikimedia projects. Wikiversity's [[Wikiversity:Mission|mission]] is to create and host a range of learning projects and learning resources. You need to find a way to contribute to this mission that doesn't involve past edits elsewhere. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:01, 22 April 2020 (UTC) == Finding Common Ground == The course [[Finding Common Ground]] is now complete. May I add it to the news myself or is there some other way to have it added? Thanks. --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 01:40, 14 March 2022 (UTC) :I've added an announcement of your course's completion to Main Page News, please modify if you wish. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 05:16, 14 March 2022 (UTC) ::Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:50, 14 March 2022 (UTC) == bug == January 5, 2023 is after December 25, 2023??? [[Special:Contributions/79.185.141.205|79.185.141.205]] ([[User talk:79.185.141.205|discuss]]) :fixed, thanks for bringing this up. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:30, 6 January 2024 (UTC) ==noinclude== {{ping|Dave Braunschweig}} I suppose, the offending <code><nowiki></noinclude></nowiki></code> on the Main Page comes from here. --[[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 21:22, 8 December 2024 (UTC) 9ncxyeyjnizanxgliffovju6vghl7qk 2690910 2690902 2024-12-08T21:45:20Z Atcovi 276019 /* noinclude */ Reply 2690910 wikitext text/x-wiki === Semi === Should this page be full protected? Obviously all the custodians should have it watchlisted, but no one other than a custodian has ever edited it. [[User:Salmon of Doubt|Salmon of Doubt]] 11:44, 13 September 2008 (UTC) * The idea of the complex protection system on the various main page templates was to leave this particular page semi-protected only so that ordinary users can add news items. This page should only be fully protected if it becomes a repeated target for vandalism, which is not yet the case. --[[User:McCormack|McCormack]] 12:07, 13 September 2008 (UTC) == Possible use of dynamicpagelist == {{#invoke:DynamicPageList |show |category=Completed resources |namespace= |count=10 |mode=unordered |ordermethod=categoryaddlastedit |orderdate=categoryaddlastdate |order=descending }} --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:19, 16 October 2016 (UTC) So, if I follow current custom, [[Main Page/News]] would appear something like: '''2016''' * '''15 October:''' The new courses [[IT Fundamentals]] and [[Assessing Human Rights]] are completed and ready for students and professionals * '''15 October:''' The new book chapter [[Motivation and emotion/Book/2016/Emotional hijacking|Emotional hijacking]] is ready for use by psychologists and their students * '''12 October:''' The Python Programming lessons [[Python Programming/Modules|Modules]] and [[Python Programming/Classes|Classes]] are completed and ready for students * '''15 August:''' '''Wikiversity celebrates its 10-year anniversary!''' * '''15 August:''' The fall semester begins for the course [[principles of radiation astronomy]]. * '''8 August:''' The [[ENG 099|open course in conversational American English]] for EFL/ESL/ELL/ESOL students starts. * '''31 July:''' For [[Wikiversity:Year of Science 2016|'''The Wikimedia Year of Science''']], try using the May 9, 2016 [[w:Transit of Mercury]] to improve the [[Stars/Sun/Locating the Sun|location of the Sun]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:22, 16 October 2016 (UTC) ::I'm confused. You wrote that "It would be great to have five to six things to put in the news each week instead of five to six for three to four months!" Are you also wanting to maintain this manually rather than using DynamicPageList to do the updates automatically? If you are working manually, there is no reason to list IT Fundamentals again, as it is only on the DynamicPageList due to vandalism. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:40, 16 October 2016 (UTC) :::Originally I was intending to try to maintain this subpage manually. Your use of dynamicpagelist suggests that at least this part could be automated. But there appears to be no easy way to include what the resources are, their approximate dates of completion and intended targets. I also wanted to include those News items that are not in that specific category. Combining the two in an automated way would really be great! --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 02:11, 17 October 2016 (UTC) ==Earlier unannounced completed resources== These would currently have been updated or improved: {{div col|colwidth=12em}} {{#invoke:DynamicPageList |show |category=Completed resources |namespace= |count=10 |mode=unordered |ordermethod=categoryaddlastedit |orderdate=categoryaddlastdate |order=descending }} {{Div col end}} --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:54, 25 June 2017 (UTC) # [[Mathematical Properties]] was announced on 20 January 2018, # [[Internet Protocol Analysis]] was announced on 15 March 2019, # [[Overcoming Hate]] was announced on 9 July 2019, # [[Unleashing Creativity]] was announced on 2 July 2019, # [[Candor]] was announced on 28 June 2019, # [[Exploring Social Constructs]] was announced on 31 May 2019, # [[Applied Programming]] was announced on 17 May 2018, # [[Finding Courage]] was announced on 17 April 2019, # [[Moral Reasoning]] was announced on 14 April 2019, # [[Coping with Ego]] was announced on 17 March 2019, # [[Resolving Anger]] was announced on 1 February 2019, # [[Communicating Power]] was announced on 29 January 2019, # [[What you can change and what you cannot]] was announced on 28 January 2019, # [[Recognizing Emotions]] was announced on 25 January 2019, # [[Appraising Emotional Responses]] was announced on 24 January 2019, and # [[Open Source 3-D Printing]] was announced on 20 January 2019. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 01:35, 15 September 2019 (UTC) == COVID-19 COBOL programmer shortage? == <blockquote>COVID-19 COBOL programmer shortage</blockquote> Where can I find information about this topic (from 12 April news item)? Thanks in advance. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 19:13, 21 April 2020 (UTC) :{{Ping|Ottawahitech}} There's quite a bit you can learn with a search engine. Just from my cursory following of the news, I know that the problem is particularly acute in New Jersey in the banking industry. If you're interested in learning, you may want to track down *How to Learn COBOL in 21 Days*, which is free online. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:28, 21 April 2020 (UTC) :{{At|Ottawahitech}} A long list of additional resources are included at [[Talk:COBOL]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:29, 21 April 2020 (UTC) ::{{At|Koavf|Dave Braunschweig}} actually there is a story behind my question. Back in 2011 I was trying to insert information into a Bank of America article and was meeting a [https://en.wikipedia.org/w/index.php?oldid=436375222&title=Talk%3ABank_of_America#The_suicide_of_Kevin_Flanagan lot of resistance], even though at the time the information was documented on another Wikipedia article where I first learned about it. The information was about a computer programmer who committed suicide in front of one of Bank of America's properties. According to the the article, he was despondent after losing his job at Bofa, which went to H1B workers. Flanagn and his layed off coleauges all of whom lost their jobs were tasked with training their replacements. ::To make a long story short, there were two persistent editors guarding information on the Bofa article, and I eventually moved on to other areas. Those two kept the story out of wikipedia, and months or even years later I found out that the article about Flanagan himself disappreaed as well. Since then I am skeptical about claims of a shortage of skilled people made by employers without the corresponding verifiable reliable sources showing a shortage actually exists. Maybe that's just me? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 02:24, 22 April 2020 (UTC) :::{{ping|Ottawahitech}} Well, that certainly escalated quickly. Without getting into the specifics of any editing disputes from the past or possible criminal disappearances, etc. it seems like your main question is, "Are there ''really'' a shortage of COBOL specialists?" and the answer is 100% yes. This has been a known problem for well over a decade (likely longer) and if you want evidence of that, I would recommend looking at job listings to see how much a company will pay someone with COBOL knowledge versus (e.g.) Python or C+. Having proficiency and especially experience in COBOL is a good way to make money in programming. —[[User:Koavf|Justin (<span style="color:grey">ko'''a'''vf</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 02:37, 22 April 2020 (UTC) :::{{At|Ottawahitech}} I'm sorry, but for me this approach crosses the line. I have recommended that you move on from past incidents on other Wikimedia projects.[https://en.wikiversity.org/w/index.php?title=User_talk%3AOttawahitech&type=revision&diff=2141168&oldid=2137899] I have strongly encouraged you to move on from past incidents on other Wikimedia projects. I am now warning you to move on from past incidents on other Wikimedia projects. Wikiversity's [[Wikiversity:Mission|mission]] is to create and host a range of learning projects and learning resources. You need to find a way to contribute to this mission that doesn't involve past edits elsewhere. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:01, 22 April 2020 (UTC) == Finding Common Ground == The course [[Finding Common Ground]] is now complete. May I add it to the news myself or is there some other way to have it added? Thanks. --[[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 01:40, 14 March 2022 (UTC) :I've added an announcement of your course's completion to Main Page News, please modify if you wish. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 05:16, 14 March 2022 (UTC) ::Thanks! [[User:Lbeaumont|Lbeaumont]] ([[User talk:Lbeaumont|discuss]] • [[Special:Contributions/Lbeaumont|contribs]]) 11:50, 14 March 2022 (UTC) == bug == January 5, 2023 is after December 25, 2023??? [[Special:Contributions/79.185.141.205|79.185.141.205]] ([[User talk:79.185.141.205|discuss]]) :fixed, thanks for bringing this up. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 22:30, 6 January 2024 (UTC) ==noinclude== {{ping|Dave Braunschweig}} I suppose, the offending <code><nowiki></noinclude></nowiki></code> on the Main Page comes from here. --[[User:Watchduck|Watchduck]] <small>([[User talk:Watchduck|quack]])</small> 21:22, 8 December 2024 (UTC) :Thanks for the watchful eye {{ping|Watchduck}} I've fixed it up. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 December 2024 (UTC) 9en957kxwuv5wvkpzl7evuefq3l19xu 0 113381 2690982 2690735 2024-12-09T07:20:52Z Bert Niehaus 2387134 /* Chapter 4 - Curves and Line Integrals */ 2690982 wikitext text/x-wiki '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_Function|Wikipedia: Holomorphic functions]] ** [[w:en:Integral_calculus|Wikipedia: Integral calculus]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity and limits|Wikipedia: Continuity and limits]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> rn7m9srbqrt1tfnlrmvzh331mw61tny 2690996 2690982 2024-12-09T10:56:44Z Bert Niehaus 2387134 /* Chapter 4 - Curves and Line Integrals */ 2690996 wikitext text/x-wiki '''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level. ==Articles== * [[Algebra II]] * [[Dummy variable]] * [[Materials Science and Engineering/Equations/Quantum Mechanics]] == Slides for Lectures == === Chapter 1 - Intoduction === * '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Numbers/From%20real%20to%20complex%20numbers&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Complex%20Numbers&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]] * '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Riemann%20sphere&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Riemann%20sphere&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Exponentiation_and_square_root&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Exponentiation_and_square_root&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 2 - Topological Foundations === * '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Sequences%20and%20series&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Sequences%20and%20series&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[/Power series/]] * '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=topological%20algebra&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] * '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Norms,%20metrics,%20topology&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Norms,%20metrics,%20topology&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 3 - Complex Derivative === * '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Holomorphic%20function&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Holomorphic%20function&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Partial%20derivative&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Partial%20Derivative&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Application%20of%20Cauchy-Riemann%20Equations&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] === Chapter 4 - Curves and Line Integrals === * '''[[Line integral|Line integral in <math>\mathbb{R}^n</math>]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Line%20integral&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Line%20integral&coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Complex_Analysis/Curves|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex_Analysis/Curves&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Holomorphic_Function|Wikipedia: Holomorphic functions]] ** [[w:en:Integral_calculus|Wikipedia: Integral calculus]] * '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Course:Complex_Analysis/Paths&author=Complex_Analysis&language=en&audioslide=yes&shorttitle=Paths&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] * '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Path_Integral&author=Course:Complex_Analysis&language=en&audioslide=yes&shorttitle=Path_Integral&coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]] ** [[w:en:Curve integral |Wikipedia: Curve integral]] ** [[w:en:Continuity and limits|Wikipedia: Continuity and limits]] ==Lectures== * [[/Cauchy-Riemann equations/]] * [[Cauchy Theorem for a triangle]] * [[Complex analytic function]] * [[Complex Numbers]] * [[Divergent series]] * [[Estimation lemma]] * [[Fourier series]] * [[Fourier transform]] * [[Fourier transforms]] * [[Laplace transform]] * [[Riemann hypothesis]] * [[The Real and Complex Number System]] * [[Warping functions]] ==Sample exams== [[/Sample Midterm Exam 1/]] [[/Sample Midterm Exam 2/]] ==See also== * [[Boundary Value Problems]] * [[Introduction to Elasticity]] * [[The Prime Sequence Problem]] * [[Wikipedia: Complex analysis]] *[[Complex number]] [[Category:Complex analysis| ]] [[Category:Mathematics courses]] [[Category:Mathematics]] <noinclude> [[de:Kurs:Funktionentheorie]] </noinclude> 03ntec5b8o6pnr65n5oowdkz1d44jd0 0 116817 2690869 1598680 2024-12-08T15:04:58Z 45.5.137.249 /* Everyday Purity */ Fix punctuation. 2690869 wikitext text/x-wiki ==Introduction:== {{TOC right | limit|limit=1}} Purity is the virtue of benevolence—acting without any trace of evil or selfish motives. Intentions are pure only when they are free of self-interest, egoism, desire, envy, cruelty, spite, greed, malice, lust, trickery, and dishonesty. Motives are pure only when they are free of power, control, and coercion.<ref> {{cite book |title=A Small Treatise on the Great Virtues: The Uses of Philosophy in Everyday Life |last=Comte-Sponville |first=André |year=2002 |publisher=Picador |isbn=978-0805045567 |pages=368}} </ref> [[w:Psychological manipulation|Manipulation]] and [[w:Exploitation|exploitation]] arise from impure motives. Pure love is giving, not taking, not a transaction, not a means to an end. Purity can only be genuine, never faked. ==The Virtue of Purity== Purity is good because evil is not. ==Everyday Purity== Examine your motives. Strive for purity each day in these ways: * Examine your speech. Are you being [[Virtues/Simplicity|simple]], straightforward, and candid, or is there some intent to deceive? Are you promoting both truth and grace? * Examine your friendships. Are the relationships ''instrumental''—sustained by the possibility of gaining some future ego or material benefit—or are they based purely on caring? Do you trade on your friendships? * Examine your gift giving and charity. Do you ever give with the thought of receiving something in return, or is your giving motivated purely by caring, kindness, [[Virtues/Generosity|generosity]], [[Virtues/Compassion|compassion]], and love? Do you ever give complements with the hope of receiving a complement in return? Do you use flattery for personal gain? * Examine your persuasion, influences, and charm. Do you ever intend to alter someone’s beliefs to align them with your political or religious ideology, rather than encouraging a broader examination of what is? Would you even be having this conversation if you were not trying to sell something? * Become aware of your own hypocrisy. Notice when you are inconsistent in what you say and what you do. Strive to increase your [[Virtues/Fidelity|fidelity]]. Beware; those who espouse purity the loudest may be the least pure. Look first to yourself. * Identify your [[w:Vice|vices]], these may include: vanity, envy, jealousy, anger, greed, overindulgence, smoking, excessive drinking, gambling, infidelity, and others. Strive to end these. * Reflect on your own sexual behaviors. Have all participants expressed their informed, adult, consent? Is anyone ever deceived, tricked, coerced, shamed, harmed, or regretful as a result? Do the behaviors bring only enjoyment to each participant? Does the sex express caring? * Examine your love. Is it conditional—sometimes intentionally withheld to coerce another—or is it unconditional? Be gentle with yourself, purity is a virtue but absolute purity is unattainable. ==Assignment== '''Part 1:''' Choose one of the areas described above to examine in detail. '''Part 2:''' For one week, for the area chosen to study, as you act keep a record of each impurity that has crept into your motives. '''Part 3:''' Resolve to purge that impurity from yourself. ==References== <references/> ==Further Reading== Students interested in learning more about the virtues of purity may be interested in the following materials: * {{cite book |title=The Righteous Mind: Why Good People Are Divided by Politics and Religion |last=Haidt |first=Jonathan |authorlink=w:Jonathan_Haidt |year=2012 |publisher=Pantheon |isbn=978-0307377906 |pages=448 }} * (Evaluate the book: ''Purity and Danger: An Analysis of Concepts of Pollution and Taboo'' ) {{Virtues}} {{CourseCat}} ck6smq6slbu3aw25pxbf94k7dra6jto 0 125529 2690987 2303187 2024-12-09T08:40:58Z RockTransport 2992610 /* Mobile Web Applications */ Mentioned the same topic two times 2690987 wikitext text/x-wiki {{it}} {{tertiary}} {{nonformal}} {{launch}} ==Mobile Web Applications== The best way to learn to develop mobile web applications is to develop web pages. This philosophy is how this course has been designed and built. It treats learning web development more like learning an art than a science. And when attending an art class the opportunity to jump right in starts usually on the first day, during the first hour of class. ==Considerations== # When on the Internet users prefer to use web apps rather than mobile apps on their mobile devices. # Learners using this learning resource should find or create their own development environment capable of hosting the technologies used within this course. A review of my series of blog posts describing the setup of a rackspace cloud server can help here; http://criticaltechnology.blogspot.com/search/label/hosting # This course will be taught more as an ART course than a SCIENCE course. So be prepared to get dirty and learn with reckless abandon. # The creation of this course / book is as an example toward [[User:Peterrawsthorne/ALD|Agile Learner Design]] ==Conventions== # Each chapter will focus only on what it needs to meet its objective. Within any feature of technology there is often many more attributes than need to be discussed to meet a chapters objectives. It is preferred to only focus on the immediate need, rather than all that is possible with any given feature. # [[Wikipedia:Pseudocode|Pseudocode]] will be used to describe all algorithms / programming logic # Each chapter will include a '''business value''' section to describe the value of the technology will have to the business. This convention is used for two reasons; first, to get away from technology for technologies sake. And second, to provide a laypersons view into why the technology is important. The business value section answers the c-level question, "What's the value of doing this?" ==Course concept map== This concept map shows the content covered by this resource. The concept map is a work in progress as it will be added to as the course continues as suggested by the [http://criticaltechnology.blogspot.com/2011/11/agile-instructional-design.html Agile Instructional Design] methodology. ==Table of Contents== * ch01. [[/Ch01|Building a Mobile HTML5 page]] – Start by building a working mobile view * ch02. [[/Ch02|MVC Three-Tier Architecture]] – The Architecture Described * ch03. [[/Ch03|Building the Model and Controller]] - Bringing context to the web page * ch04. [[/Ch04|Information Architecture]] – Designing the user experience * ch05. [[/Ch05|Touching the business tier]] – Thinking about services * apxA. [[/ApxA|Debugging with firebug]] - fixing and improving your code ==Related Badges== [[MWA/Badges|Mobile Web Application Badges]] - a set of badges awarded for demonstrating mastery of the different sections within this book. [[Category:Computer programming]] [[Category:Art and Design]] otk4v27txpaz9smxrcz6f6l5rp19v1z 0 203942 2690948 2690225 2024-12-09T03:19:31Z Young1lim 21186 /* Lambda Calculus */ 2690948 wikitext text/x-wiki ==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]]) </br> ==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]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==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]]) </br> </br> ==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.20241205.pdf |pdf]]) </br> </br> ==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 & 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 & 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]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> 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] [[Category:Haskell|programming in plain view]] ccykeircsf1lydswqthtw8euuq0k9ra 2690950 2690948 2024-12-09T03:21:11Z Young1lim 21186 /* Lambda Calculus */ 2690950 wikitext text/x-wiki ==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]]) </br> ==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]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==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]]) </br> </br> ==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.20241206.pdf |pdf]]) </br> </br> ==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 & 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 & 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]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> 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] [[Category:Haskell|programming in plain view]] 5h2xniqorhq6yppvshfvjlxp5rwv6hx 2690952 2690950 2024-12-09T03:21:56Z Young1lim 21186 /* Lambda Calculus */ 2690952 wikitext text/x-wiki ==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]]) </br> ==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]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==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]]) </br> </br> ==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.20241207.pdf |pdf]]) </br> </br> ==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 & 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 & 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]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> 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] [[Category:Haskell|programming in plain view]] mfgfk0fbgsjyeb9x7ru7ysm5xvlryqm 2690954 2690952 2024-12-09T03:22:51Z Young1lim 21186 /* Lambda Calculus */ 2690954 wikitext text/x-wiki ==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]]) </br> ==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]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==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]]) </br> </br> ==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.20241209.pdf |pdf]]) </br> </br> ==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 & 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 & 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]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> 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] [[Category:Haskell|programming in plain view]] arh6jlnoyu2l4kbipnndrfx0iv3dlql 1 205923 2690986 2392112 2024-12-09T08:37:42Z RockTransport 2992610 /* Very nice quote. */ new section 2690986 wikitext text/x-wiki <blockquote>'''''"Knowledge grows when shared."'''''<br />-[[w:Bhartrihari|Bhartrihari]]</blockquote> == Discuss == I have not been able to find a source for this quote. Does anyone know where this was published? --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 14:25, 12 September 2009 (UTC) Bhartrihari was an Indian poet. This might be an excerpt from one of his works, a collection of poems called 'bhartrihari subhashitalu' or ' subhashita ratnalu '.{{unsigned2|06:18, 14 May 2012|122.170.80.93}} [[Category:Quotes]] == Computer == Can you create a application on any computer and how ? [[Special:Contributions/102.68.120.90|102.68.120.90]] ([[User talk:102.68.120.90|discuss]]) 17:04, 2 May 2022 (UTC) == Very nice quote. == This is a very good quote used to increase engagement of learning. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:37, 9 December 2024 (UTC) ps324vyd4pw4qrft3wqjuktrg1xfown 0 212733 2690968 2690670 2024-12-09T04:32:43Z Young1lim 21186 /* Using Libraries */ 2690968 wikitext text/x-wiki ==''' Part I '''== <!----------------------------------------------------------------------> === Introduction === * Overview * Memory * Number <!----------------------------------------------------------------------> === 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 <!----------------------------------------------------------------------> === 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.20241209.pdf |pdf]]) * Namespaces ([[Media:Python.Work2.Scope.1A.20231021.pdf |pdf]]) <!----------------------------------------------------------------------> === Handling Repetition === * Control ([[Media:Python.Repeat1.Control.1.A.20230314.pdf |pdf]]) * Loop ([[Media:Repeat2.Loop.1A.20230401.pdf |pdf]]) <!----------------------------------------------------------------------> === 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]]) <!----------------------------------------------------------------------> === 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]]) <!----------------------------------------------------------------------> === Handling Various Kinds of Data === * Types * Operators ([[Media:Python.Data3.Operators.1.A.pdf |pdf]]) * Files ([[Media:Python.Data4.File.1.A.pdf |pdf]]) <!----------------------------------------------------------------------> === Class and Objects === * Classes & Objects ([[Media:Python.Work2.Class.1A.20230906.pdf |pdf]]) * Inheritance <!----------------------------------------------------------------------> </br> == Python in Numerical Analysis == </br> </br> 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] jgbh1ul94hixhx4v5e9ljuayxq921es 0 232469 2690965 2690662 2024-12-09T04:14:15Z Young1lim 21186 /* ARM Assembly Programming (II) */ 2690965 wikitext text/x-wiki == '''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]] </br> == '''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 </br> == '''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 </br> == '''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.20241208.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]]) </br> * 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]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''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 </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''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]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] c3uf8ket45s1bdrpk7sja01zwrdlywt 2690966 2690965 2024-12-09T04:15:06Z Young1lim 21186 /* ARM Assembly Programming (II) */ 2690966 wikitext text/x-wiki == '''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]] </br> == '''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 </br> == '''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 </br> == '''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.20241209.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]]) </br> * 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]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''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 </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''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]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] i2qjfgvjk1kw7mis091faigjtw42lu6 0 263813 2690913 2690737 2024-12-08T21:48:26Z Scogdill 1331941 2690913 wikitext text/x-wiki ==Also Known As== * Family name: The Hon. Algernon Bourke * Lady Florence Bourke * See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the family the Hon. Algernon Bourke was a member of. == 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,<ref name=":6">1911 England Census.</ref> he seems to have spent much of his adult life generally in England and especially in London. Lady Florence Bourke was a noted horsewoman, exhibited at dog shows and was "an appreciative listener to good music."<ref>''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.</ref> It is not clear that the Hon. Algernon Bourke and Mr. Algernon were or were not the same person, so this page treats them as two people until certainty can be achieved. Called 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]], "one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys"<ref name=":8">"Metropolitan Notes." ''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.</ref> * Caroline, Duchess of Montrose — her "legal advisor" on the day of her marriage to Marcus Henry Milner<ref>"Metropolitan Notes." ''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.</ref> == Organizations == === The Hon. Algernon Bourke === * Eton * Cambridge University, Trinity College, 1873, Michaelmas term<ref name=":7">Cambridge University Alumni, 1261–1900. Via Ancestry.</ref> * 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,<ref>{{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.</ref> Manager (1897)<ref>"Side Lights on Drinking." ''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.</ref> * Stock Exchange * Willis's Rooms<blockquote>... 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.<ref>"Lenten Dullness." ''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.</ref></blockquote> *The Franco-English Tunisian Esparto Fibre Supply Company, Limited, one of the directors<ref>''Money Market Review'', 20 Jan 1883 (Vol 46): 124.</ref> === Mr. Algernon Bourke === * Head, Messrs. Bourke and Sandys, "that well-known firm of stockbrokers"<ref name=":8" /> == Timeline == '''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a "convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier."<ref>{{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.</ref> 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.<ref>"The Fatal Accident to a Sheffield Student at Cambridge." ''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.</ref> '''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 "the recent death of a near relative of the bride."<ref>"Court Circular." ''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.</ref> '''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.<ref name=":12">"Hon. Caroline Agnes Horsley-Beresford." {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> According to the ''Nottingham Evening Post'' of 31 July 1888,<blockquote>LONDON GOSSIP. (From the ''World''.) The marriage of "Mr. Manton" 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. "Harry" Milner (familiarly known in the City as "Millions") 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]].<ref name=":8" /></blockquote>'''1888 December 1st week''', according to "Society Gossip" from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:<blockquote>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.<ref>"Society Gossip. What the ''World'' Says." ''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.</ref></blockquote>'''1889 – 1899 January 1''', the Hon. Algernon Bourke was "proprietor" of White's Club, St. James's Street.<ref name=":9">"The Hon. Algernon Bourke's Affairs." ''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.</ref> '''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club. '''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. '''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:<blockquote>THE HON. ALGERNON BOURKE'S AFFAIRS. 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,E00, 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.<ref name=":9" /></blockquote>'''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.<ref>"London Day by Day." ''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 & Courier'' (London). Print p. 8.</ref> == 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#"Oriental" Procession|"Oriental" procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].<ref name=":2" /><ref name=":3" /> 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 "Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô" in costume is photogravure #128 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Hon. Mrs. Algernon Bourke as Salammbo."<ref>"Mrs. Algernon Bourke as Salammbo." ''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.</ref> ==== Newspaper Accounts ==== The Hon. Mrs. A. Bourke was dressed as * Salambo in the Oriental procession.<ref name=":2">"Fancy Dress Ball at Devonshire House." ''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.</ref><ref name=":3">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref> * "(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."<ref>“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.</ref>{{rp|p. 40, Col. 3a}} *"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."<ref>“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.</ref>{{rp|p. 3, Col. 3b}} ==== Salammbô ==== Salammbô is the eponymous protagonist in Gustave Flaubert's 1862 novel.<ref name=":5">{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.</ref> Ernest Reyer's opera ''Salammbô'' was based on Flaubert's novel and published in Paris in 1890 and performed in 1892<ref>{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.</ref> (both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well<ref name=":5" />). 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 "Spy," ''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 "Hon. Algernon Henry Bourke as Izaak Walton" in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4" /> The printing on the portrait says, "The Hon. Algernon Bourke as Izaak Walton."<ref>"Hon. Algernon Bourke as Izaak Walton." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.</ref> 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''.<ref>{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.</ref> 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 "Algy," by Leslie Ward ("Spy") was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its "Men of the Day" series,<ref>{{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)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> 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.<ref name=":3" /> 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<ref>{{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&oldid=988654078|journal=Wikipedia|language=en}}</ref> *Occupation: journalist. 1895: restaurant, hotel and club owner and manager<ref>''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.</ref> === Residences === *Ireland: 1873: Palmerston House, Straffan, Co. Kildare.<ref name=":7" /> Not Co. Mayo? *1890: 33 Cadogan Terrace *1891: 33 Cadogan Terrace, Kensington and Chelsea, a dwelling house<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.</ref> *1894: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.</ref> *1900: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.</ref> *1911: 1911 Fulham, London<ref name=":6" /> *20 Eaton Square, S.W. (in 1897)<ref name=":0">{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. & C. Black|language=en}} 712, Col. 1b.</ref> (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]]) == Family == *Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)<ref>"Hon. Algernon Henry Bourke." {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> *Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)<ref name=":1">"Guendoline Irene Emily Stanley." {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> #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.<ref name=":0" /> ==== Other Bourkes ==== *Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)<ref>"Hubert Edward Madden Bourke-Borrowes." {{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.</ref> *Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;<ref>"Lady Eva Constance Aline Bourke." {{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.</ref> he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926. == Writings == * Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892. == 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 "Oriental" 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#"Oriental" Procession|"Oriental" 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 == {{reflist}} 947kik241hgoqn4zx3pgd5dybv2ja44 2690914 2690913 2024-12-08T21:50:49Z Scogdill 1331941 2690914 wikitext text/x-wiki ==Also Known As== * Family name: The Hon. Algernon Bourke * Lady Florence Bourke * See also the [[Social Victorians/People/Mayo|page for the Earl of Mayo]], the family the Hon. Algernon Bourke was a member of. == 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,<ref name=":6">1911 England Census.</ref> he seems to have spent much of his adult life generally in England and especially in London. Lady Florence Bourke was a noted horsewoman, exhibited at dog shows and was "an appreciative listener to good music."<ref>"Vanity Fair." ''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.</ref> It is not clear that the Hon. Algernon Bourke and Mr. Algernon were or were not the same person, so this page treats them as two people until certainty can be achieved. Called 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]], "one of the zealous assistants of that well-known firm of stockbrokers, Messrs. Bourke and Sandys"<ref name=":8">"Metropolitan Notes." ''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.</ref> * Caroline, Duchess of Montrose — her "legal advisor" on the day of her marriage to Marcus Henry Milner<ref>"Metropolitan Notes." ''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.</ref> == Organizations == === The Hon. Algernon Bourke === * Eton * Cambridge University, Trinity College, 1873, Michaelmas term<ref name=":7">Cambridge University Alumni, 1261–1900. Via Ancestry.</ref> * 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,<ref>{{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.</ref> Manager (1897)<ref>"Side Lights on Drinking." ''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.</ref> * Stock Exchange * Willis's Rooms<blockquote>... 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.<ref>"Lenten Dullness." ''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.</ref></blockquote> *The Franco-English Tunisian Esparto Fibre Supply Company, Limited, one of the directors<ref>''Money Market Review'', 20 Jan 1883 (Vol 46): 124.</ref> === Mr. Algernon Bourke === * Head, Messrs. Bourke and Sandys, "that well-known firm of stockbrokers"<ref name=":8" /> == Timeline == '''1872 February 8''', Richard Bourke, 6th Earl of Mayo was assassinated while inspecting a "convict settlement at Port Blair in the Andaman Islands ... by Sher Ali Afridi, a former Afghan soldier."<ref>{{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.</ref> 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.<ref>"The Fatal Accident to a Sheffield Student at Cambridge." ''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.</ref> '''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 "the recent death of a near relative of the bride."<ref>"Court Circular." ''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.</ref> '''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.<ref name=":12">"Hon. Caroline Agnes Horsley-Beresford." {{Cite web|url=https://thepeerage.com/p6863.htm#i68622|title=Person Page|website=thepeerage.com|access-date=2020-11-21}}</ref> According to the ''Nottingham Evening Post'' of 31 July 1888,<blockquote>LONDON GOSSIP. (From the ''World''.) The marriage of "Mr. Manton" 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. "Harry" Milner (familiarly known in the City as "Millions") 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]].<ref name=":8" /></blockquote>'''1888 December 1st week''', according to "Society Gossip" from the ''World'', the Hon. Algernon Bourke was suffering from malaria, presumably which he caught when he was in South Africa:<blockquote>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.<ref>"Society Gossip. What the ''World'' Says." ''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.</ref></blockquote>'''1889 – 1899 January 1''', the Hon. Algernon Bourke was "proprietor" of White's Club, St. James's Street.<ref name=":9">"The Hon. Algernon Bourke's Affairs." ''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.</ref> '''1892''', the Hon. Algernon Bourke privately published his ''The History of White's'', the exclusive gentleman's club. '''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. '''1899 October 19, Thursday''', the Hon. Algernon Bourke had a bankruptcy hearing:<blockquote>THE HON. ALGERNON BOURKE'S AFFAIRS. 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,E00, 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.<ref name=":9" /></blockquote>'''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.<ref>"London Day by Day." ''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 & Courier'' (London). Print p. 8.</ref> == 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#"Oriental" Procession|"Oriental" procession]] at the [[Social Victorians/1897 Fancy Dress Ball | Duchess of Devonshire's fancy-dress ball]].<ref name=":2" /><ref name=":3" /> 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 "Guendoline Irene Emily Bourke (née Sloane-Stanley) as Salammbô" in costume is photogravure #128 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Hon. Mrs. Algernon Bourke as Salammbo."<ref>"Mrs. Algernon Bourke as Salammbo." ''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.</ref> ==== Newspaper Accounts ==== The Hon. Mrs. A. Bourke was dressed as * Salambo in the Oriental procession.<ref name=":2">"Fancy Dress Ball at Devonshire House." ''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.</ref><ref name=":3">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref> * "(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."<ref>“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.</ref>{{rp|p. 40, Col. 3a}} *"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."<ref>“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.</ref>{{rp|p. 3, Col. 3b}} ==== Salammbô ==== Salammbô is the eponymous protagonist in Gustave Flaubert's 1862 novel.<ref name=":5">{{Cite journal|date=2024-04-29|title=Salammbô|url=https://en.wikipedia.org/w/index.php?title=Salammb%C3%B4&oldid=1221352216|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Salammb%C3%B4.</ref> Ernest Reyer's opera ''Salammbô'' was based on Flaubert's novel and published in Paris in 1890 and performed in 1892<ref>{{Cite journal|date=2024-04-11|title=Ernest Reyer|url=https://en.wikipedia.org/w/index.php?title=Ernest_Reyer&oldid=1218353215|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Ernest_Reyer.</ref> (both Modest Mussorgsky and Sergei Rachmaninoff had attempted but not completed operas based on the novel as well<ref name=":5" />). 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 "Spy," ''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 "Hon. Algernon Henry Bourke as Izaak Walton" in costume is photogravure #129 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":4" /> The printing on the portrait says, "The Hon. Algernon Bourke as Izaak Walton."<ref>"Hon. Algernon Bourke as Izaak Walton." ''Diamond Jubilee Fancy Dress Ball''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158492/Hon-Algernon-Henry-Bourke-as-Izaak-Walton.</ref> 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''.<ref>{{Cite journal|date=2021-09-15|title=Izaak Walton|url=https://en.wikipedia.org/w/index.php?title=Izaak_Walton&oldid=1044447858|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Izaak_Walton.</ref> 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 "Algy," by Leslie Ward ("Spy") was published in the 20 January 1898 issue of ''Vanity Fair'' as Number 702 in its "Men of the Day" series,<ref>{{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)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> 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.<ref name=":3" /> 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<ref>{{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&oldid=988654078|journal=Wikipedia|language=en}}</ref> *Occupation: journalist. 1895: restaurant, hotel and club owner and manager<ref>''Cheltenham Looker-On'', 23 March 1895. Via Ancestry but taken from the BNA.</ref> === Residences === *Ireland: 1873: Palmerston House, Straffan, Co. Kildare.<ref name=":7" /> Not Co. Mayo? *1890: 33 Cadogan Terrace *1891: 33 Cadogan Terrace, Kensington and Chelsea, a dwelling house<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970, Register of Voters, 1891.</ref> *1894: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1894. Via Ancestry.</ref> *1900: 181 Pavilion Road, Kensington and Chelsea<ref>Kensington and Chelsea, London, England, Electoral Registers, 1889–1970. Register of Voters, 1900. Via Ancestry.</ref> *1911: 1911 Fulham, London<ref name=":6" /> *20 Eaton Square, S.W. (in 1897)<ref name=":0">{{Cite book|url=https://books.google.com/books?id=Pl0oAAAAYAAJ|title=Who's who|date=1897|publisher=A. & C. Black|language=en}} 712, Col. 1b.</ref> (London home of the [[Social Victorians/People/Mayo|Earl of Mayo]]) == Family == *Hon. Algernon Henry Bourke (31 December 1854 – 7 April 1922)<ref>"Hon. Algernon Henry Bourke." {{Cite web|url=https://www.thepeerage.com/p29657.htm#i296561|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> *Guendoline Irene Emily Sloane-Stanley Bourke (c. 1869 – 30 December 1967)<ref name=":1">"Guendoline Irene Emily Stanley." {{Cite web|url=https://www.thepeerage.com/p51525.htm#i515247|title=Person Page|website=www.thepeerage.com|access-date=2020-12-10}}</ref> #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.<ref name=":0" /> ==== Other Bourkes ==== *Hubert Edward Madden Bourke (after 1925, Bourke-Borrowes)<ref>"Hubert Edward Madden Bourke-Borrowes." {{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.</ref> *Lady Eva Constance Aline Bourke, who married [[Social Victorians/People/Dunraven|Windham Henry Wyndham-Quin]] on 7 July 1885;<ref>"Lady Eva Constance Aline Bourke." {{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.</ref> he became 5th Earl of Dunraven and Mount-Earl on 14 June 1926. == Writings == * Bourke, the Hon. Algernon. ''The History of White's''. London: Algernon Bourke [privately published], 1892. == 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 "Oriental" 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#"Oriental" Procession|"Oriental" 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 == {{reflist}} b1khw8phs4e2jf0ym023nsxoaunckoj 0 264335 2690912 2689051 2024-12-08T21:48:19Z Scogdill 1331941 2690912 wikitext text/x-wiki = Newspapers and Magazines = See also the page collecting [[Social Victorians/People/Working in Publishing|people who worked in publishing and journalism]]: publishers, journalists (including "[[Social Victorians/People/Working in Publishing#Journalists|Aristocratic Lady Journalists]]"), illustrators, editors, proprietors, and so on. Magazines and less-frequently published periodicals are [[Social Victorians/Newspapers#Magazines and Other Periodicals|later on this page]]. == Periodicals That Published Society and London Gossip (Mitchell's) == *The Argus *The Bookman *[[Social Victorians/Newspapers#The Court Journal: Gazette of the Fashionable World, Literature, Music and the Fine Arts|The Court Journal: Gazette of the Fashionable World, Literature, Music and the Fine Arts]] *[[Social Victorians/Newspapers#The (London) Echo|The (London) Echo]] *[[Social Victorians/Newspapers#Fashionable London: An Illustrated Journal for Ladies|Fashionable London: An Illustrated Journal for Ladies]] *[[Social Victorians/Newspapers#The Gentlewoman: The Illustrated Weekly Journal for Gentlewomen|The Gentlewoman: The Illustrated Weekly Journal for Gentlewomen]] *Hampshire Telegraph and Sussex Chronicle *The Isle of Wight Guardian *[[Social Victorians/Newspapers#The Lady|The Lady]] *[[Social Victorians/Newspapers#Lady of the House|The Lady of the House]] *The Lady's Magazine (La Moniteur de la Mode) [about class rather than gossip] *The Lady's World (see [[Social Victorians/Newspapers#The Woman's World|The Woman's World]]) *The Licensed Victualler's Mirror *Observer *The Owl *The People *[[Social Victorians/Newspapers#The Queen, The Lady's Newspaper|The Queen, the Lady's Newspaper]] *[[Social Victorians/Newspapers#The Sketch: A Journal of Art and Actuality|The Sketch: A Journal of Art and Actuality]] *[[Social Victorians/Newspapers#Society|Society]] *St. James's Budget *[[Social Victorians/Newspapers#The St. James's Gazette|The St. James's Gazette]] *The Stage *[[Social Victorians/Newspapers#Vanity Fair|Vanity Fair]] *Warder and Dublin Weekly Mail *Waverley *The Weekly Sun *The Western Weekly Mercury *Whitehall Review *[[Social Victorians/Newspapers#The Woman's World|The Woman's World]] *Wrexham Argus and North Wales Athlete The Central Press, a press agency, says it provides "Lobby Gossip" (Mitchell's 188) and "Society Gossip" (Mitchell's 304). == Papers from Outside the U.K. That Played a Role == *''The Beacon'' (in Poona, India) *''Mercure de France'' *''Overland Mail'' (written for India; special edition for China) *The New York ''Herald'' (9 March 1858–31 January 1920; British Library DSC Shelfmark 6089.303000n) *The Paris ''Temps'' (British Library DSC Shelfmark 8790.050000) == Other Newspapers == * [[Social Victorians/Newspapers#The Belfast News-Letter|The Belfast News-Letter]] * ''The Echo (''1868–) (British Library DSC Shelfmark 3647.367450n) * ''The Glasgow Herald'' (26 August 1805–) * [[Social Victorians/Newspapers#Lloyd's Weekly Newspaper|Lloyd's Weekly Newspaper]] * [[Social Victorians/Newspapers#The London Daily News|The London Daily News]] * London ''Daily Telegraph'' (1855–),<blockquote>founded by Joseph Moses Levy in a market in which there were ten newspapers, so he made his paper less expensive than the rest. Very quickly it was outselling the ''Times.'' In its early days, under the editorship of Levy and his employees, the paper supported liberal causes and governmental reform. It also sensationalized its stories. Some headlines from the 1850s included the following: "A Child Devoured by Pigs," "Extraordinary Discovery of Man-Woman in Birmingham," "Shocking Occurrence: Five Men Smothered in a Gin Vat." In keeping with its sensationalistic approach, the paper focused on crime and court reporting. In the 1870s, the leadership on the paper was politically conservative. Edwin Arnold was editor, and he was not replaced until 1899. In the early 1880s a reporter on the paper helped solve a murder on a train. The murderer was identified by the first portrait block published in a newspaper, and he was subsequently convicted and executed. The paper would have been associated with investigative journalism. (http://www.spartacus.schoolnet.co.uk/JreynoldsN.htm; link no longer works, server gone) (ISSN 03071235. British Library DSC Shelfmark 3512.450000f)</blockquote>In 1895 ''Mitchell's Newspaper Press Directory'' says that the ''Daily Telegraph''<nowiki/>'s politics were liberal, the "Latest Time for Ads." was 7 p.m., and the "Time Published" was 5 a.m.<ref name=":2" />{{rp|56}} <blockquote>DAILY TELEGRAPH. I''d''. Established June, 20, 1855. P<small>RINCIPLES</small>: Liberal. The ''Daily Telegraph'', a morning journal which, while thoroughly devoted to the large interests of the Liberal cause, has not unfrequently taken an independent course on the merits of particular questions. Over and above its recognized political position as the popular exponent of Liberal views, it has acquired an unequalled celebrity through the promptitude, the fulness, and the variety of its telegraphic advices; the enterprise which its conductors have shown when events of great national or international interest demanded early and ample description; and the novelty and freshness of the social articles, which are a constant feature of the paper, both in its leading columns and elsewhere. The popularity and influence of the ''Daily Telegraph'' are alike very great. Published by Archibald Johnstone, 135, Fleet Street, E.C.<ref name=":2" />{{rp|59}}</blockquote> * The London ''Evening News''.<blockquote>The ''Evening News'' joined the highly competitive group of London daily newspapers in 1894 when it was purchased by journalist Alfred Harmsworth. Under Harmsworth the newspaper was successful and rather sensationalistic, with illustrations and headlines like "Was It Suicide or Apoplexy?, Another Battersea Scandal, Bones in Bishopgate, Hypnotism and Lunacy and Killed by a Grindstone" (http://www.spartacus.schoolnet.co.ul/Jevening.htm [link no longer works, server gone]). Harmsworth claimed in November 1894 that his newpaper had the largest circulation in the world — 394,447 — and that the only reason the sales were below half a million copies was the number of printing presses he owned. When his daughter was born in January 1889, [[Social Victorians/People/Arthur Conan Doyle|Arthur Conan Doyle]] put the announcement in the ''Evening News'': "CONAN DOYLE. On the 28th instant, at Bush Villa, Elm Grove, Mrs Conan Doyle, wife of A. Conan Doyle MD, of a daughter" (Stavert 136).</blockquote> * [[Social Victorians/Newspapers#London Standard and the London Evening Standard|The London Standard and Evening Standard]] * The [[Social Victorians/Newspapers#The Man of Ross|Man of Ross]] * ''The National Observer'' * ''Reynold's Weekly Newspaper'' <blockquote>had, by the end of the century, been a fixture in London journalism for many years and was, in its own words, "devoted to the cause of freedom and in the interests of the enslaved masses." Founded in 1850, it owed some of its very large circulation to its price — George William Reynolds lowered the price from 4 shillings to a penny in 1864, and by 1875 its circulation was 350,000 a week. When Reynolds died in 1894, the paper was taken over by liberal M.P. James Henry Dalziel, who "brought in several new features including a women's page, serial stories, words and music of popular songs and help finding missing relatives and friends" (http://www.spartacus.schoolnet.co.ul/JreynoldsN.htm; link no longer works, server gone).</blockquote> * ''The Scottish Leader'' (3 January 1887 – 4 July 1894?) * ''The Star'', <blockquote>founded in 1887 by politically radical journalist and Irish nationalist T. P. O'Connor. ''The Star'' hired writers for their radical beliefs. Assistant editor H. W. Massingham also hired well-known writers for their talents and names. He knew [[Social Victorians/People/George Bernard Shaw|George Bernard Shaw]] and hired him to be an assistant leader-writer. Reporter Ernest Clarke is remembered by O'Connor in his ''Memoirs'' like this: "He might be trusted to work up any sensational news of the day, and helped, with [his coverage of] Jack the Ripper, to make gigantic circulations hitherto unparalleled in evening journalism" (http://www.spartacus.schoolnet.co.uk/; link no longer works, server gone).</blockquote> * The [[Social Victorians/Newspapers#The Star of Guernsey|Star of Guernsey]] * The ''St. James's Gazette'' * [[Social Victorians/Newspapers#Westminster Gazette|Westminster Gazette]] === The Belfast News-Letter === The ''Belfast News-Letter'' began publication in 1737<ref name=":0">MJH/MaT [Matthew James Huggins/Matthew Taunton]. "Belfast News-Letter (1737–)." ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland''. Laurel Brake and Marysa Demoor, gen. eds. Gent: Academia Press; London: The British Library, 2009: 44, Col. 2b – 45, Col. 1a.</ref>{{rp|44, Col. 2b}}; by the second half of the 19th century it reported local news and "dedicated an unusual amount of column inches to literary* matters for a newspaper and printed sports'* reports, articles on horticulture and gardening*, and pieces detailing the latest developments in ladies' fashion."<ref name=":0" />{{rp|45, Col. 1a; asterisks sic, references to articles in the book}} It came out on Wednesday and Saturdays and cost 4d.<ref name=":0" />{{rp|44, Col. 2b}} ===The (London) Daily News=== In 1895 ''Mitchell's Newspaper Press Directory'' says that the ''Daily News''<nowiki/>'s politics were liberal, the "Latest Time for Ads." was 7 p.m., and the "Time Published" was 5 a.m.<ref name=":2" />{{rp|55}}. <blockquote>Daily News. 1d. Established Jan. 21, 1846. Principles: Liberal and Independent. It is very ably conducted in every department; and neither in its politics or literature, its domestic or foreign news, its English, American, or Continental correspondence and telegrams, yields the palm to any of its contemporaries. Its literary, dramatic, and musical articles are distinguished by great ability. Published by T. Britton, 19, 20, 21, Bouverie Street; (Office for Advertisements) 67, Fleet Street, W.C. (Advt. p. 32.)<ref name=":2" /> (58)</blockquote> ''Daily News'' ad in ''Mitchell's Newspaper Press Directory'', 1895: <blockquote>Daily News Office,<br> 67, Fleet Street, London.<br> 1895.<br> Important to Advertisers.<br> The Daily News<br> Has<br> The Largest Circulation<br> Of Any Liberal Paper in the World.<br> The Daily News is now the leading Liberal organ. It has the largest circulation of any liberal paper in the world, and is, therefore, the best channel for Advertisements of every description.<br> [C. Mitchell & Co., Advertising Agents and Contractors, 12 and 13, Red Lion Court, Fleet Street, London, E.C.] (32).</blockquote> The ''Daily News'' was edited by Charles Dickens early on. Editor William Black "retired from journalism" in 1876 (Brake Demoor 57 a–b). Conservative Edward Tyas Cook was editor between 1895 and 1901, when he was dismissed by the new owners, the Cadbury family. Henry Labouchere was part-proprietor beginning in 1868 (Brake Demoor 338a). According to ''The Life of Henry Labouchere'', which is quoting ''Fifty Years of Fleet Street: The Life and Recollections of Sir John Robinson'', <blockquote> Sir John Robinson thus describes the syndicate of which Mr. Labouchere became a member: "The proprietors of the Daily News, a small syndicate which never exceeded ten men, were a mixed body, hardly any two of whom had anything in common. The supreme control in the ultimate resort rested with three of them, Mr. Henry Oppenheim, the well-known financier, with politics of no very decided kind; Mr. Arnold Morley, a Right Honourable, an ex-party Whip, / and a typical ministerial Liberal; and Mr. Labouchere, the Radical, financier, freelance. Others had but a small holding, and practically did not count, save as regards any moral influence they might bring to brea on their colleagues at Board meetings."{{rp|Thorold 95–96}}</blockquote> Labouchere sold his share in 1895 (Thorold 96): <blockquote>On Mr. Gladstone's withdrawal from public life," he wrote in ''Truth'', "the party, or rather a majority of the officialdom of the party became tainted with Birmingham imperialism. My convictions did not allow me to be connected with a newspaper which supported a clique of intriguers that had captured the Liberal ship, and that accepted blindly these intriguers as the representatives of Liberalism in regard to our foreign policy.</blockquote> It looks like when Robinson stepped down, the proprietors were Oppenheim and Morley until the paper was sold to the next syndicate, which included George Cadbury{{rp|Thomas 380}}. === The (London) Echo === According to the 1895 Mitchell's, the ''Echo'' was an evening paper and in its quick overview says,<blockquote>ECHO. Daily, 1''d''. Established December, 1868. P<small>RINCIPLES</small>: Liberal Unionist. It contains, in a condensed form, all the news of the day — devoting much space to "city matters," and giving details of all "business done" on the Stock Exchange. The ''Echo'' comments fearlessly on politics and statesmen. It endeavours to promote the national welfare. It strives to secure peace, to enforce economy, and to uphold a national policy enlightened by universal education. Published at 22, Catherine Street. W.C. (Advt., p. 247.)<ref name=":2" />{{rp|60, Col. 1b}}</blockquote> The advertisement says,<blockquote>Echo. Established Quarter of a Century. <small>FAVOURITE EVENING PAPER FOR FAMILY READING</small>. Largest London Circulation. The <small>ECHO</small> is a daily newspaper and review, containing, in a condensed form, all the news of the day, in anticipation of the following day's morning paper. The <small>ECHO</small> is now acknowledged to be one of the best mediums for advertisers. In addition to its having the largest London circulation, (which on occasions reaches almost 300,000 [? the 3 is not clear]), its convenient size, and the excellent arrangement of its advertisements, ensure all the announcements appearing in its columns being brought directly under the notice of its very large number of readers. The <small>ECHO</small>, price One Halfpenny, can be obtained of any news agents in town or country, or a copy will be sent post-free to any address in the United Kingdom, at the rate of One Penny daily, viz., 26s. for twelve months; 13s. for six months; or 6.6d. for three months. P.O. Orders to be made payable to J. Passmore Edwards, 22, Catherine-st., Strand, London, W.C.<ref name=":2" />{{rp|246, Col. 2b}}</blockquote> === Fashionable London: An Illustrated Journal for Ladies === 1892–? The British Library may have a run; the Bodleian seems to as well. ===The Gentlewoman: The Illustrated Weekly Journal for Gentlewomen=== According to the 1895 Mitchell's, ''The Gentlewoman''<blockquote>Illustrated weekly newspaper for ladies, with a very Iarge and increasing circulation all over the kingdom, on the continent, in America and the colonies tinent, in America and the colonies, amongst the best and public most wealthy class.</blockquote> *According to an ad in the 1905 Newspaper Press Directory, the Gentlewoman was a weekly published on Thursday (NPD 1905 94). *It was a women's (ladies') magazine. *1890–1926 *The address was 70–76 Long Acre, London, W.C. (NPD 1905 94). *It carried illustrated interviews: <quote>the subject was often an aristocratic woman and the interview was as much about the decor and furnishings of her home as about her own achievements. These interviews blended with the advice on furnishing and house decoration which became increasingly popular feature in all kinds of magazines for women at this time. They also exploited the techniques of the new journalism to suggest an intimacy with the great and famous into whose most private rooms the reader was allowed to look</quote> (Beetham and Boardman 59). <blockquote>Gentlewoman (The). Thursday, 6d.<br> Established 1890. Illustrated weekly newspaper for ladies, with a very large and increasing circulation all over the kingdom, on the Continent, in America and the Colonies, amongst the best and most wealthy class.<br> Published at 70–76, Long Acre, W.C. (Advt., p. 96.)</blockquote>{{rp|NPD 1905 71}}. [IMG] (Who's Who 55 31) ===The Graphic=== According to the 1895 Mitchell's Newspaper Directory, ''The Graphic'' was a weekly, published on Fridays, which sold for 6d. Its description read as follows: <blockquote>Principles: Independent. An admirably illustrated journal, combining "Literary excellence with artistic beauty." The illustrations are in the first style of art. The literary portion of the paper is admirable in its arrangement, and a series of essays and notices on the topics of the day add greatly to its attractive character. Stories by popular authors appear weekly, illustrated by eminent artists.<ref name=":2" /> (68)</blockquote> It was "of small folio size (15.5in x 11.5in), with 3 cols of letterpress..., featuring at least 20 engravings mainly of larger size."<ref name=":6">Law, Graham. "The Illustrated London News and The Graphic." ''The Illustrated London News (1842-1901) and The Graphic (1869-1901)''. Retrieved September 2023. https://victorianfictionresearchguides.org/the-illustrated-london-news-and-the-graphic/.</ref> By the late 1890s, it had grown to 32 pages and had a number of supplements.<ref name=":6" /> The ''Graphic'' had a ladies' column in the 1890s and 1900s written by Lady Violet Greville, "Place aux Dames":<blockquote>Lady Violet claimed, when offered the ''Graphic'' job, that all her suggestions for subject-matter — art, literature, theatre, dress — were rejected on the grounds that they already had writers for those topics — and she should just write whatever she liked! She clearly did, earning the compliment from fellow journalist Mary Billington, (who eventually ran the "women's department" at the ''Daily Telegraph'') that as a writer she combined "daring, brilliancy, and romance":. In particular she championed the cause of sports for women.<ref name=":5" /></blockquote>See the paragraph under the ''[[Social Victorians/Newspapers#The Illustrated London News|Illustrated London News]]'' about Florence Fenwick-Miller and Violet Greville's roles in articulating the subtle differences between the ''Graphic'' and the ''Illustrated London News'' on the topic of the New Woman. ==== Proprietors, Publishers, Printers, Editors ==== William Luson Thomas was Managing Director between 1869 and 1900.<ref name=":6" /> E. J. Mansfield at 190, Strand, was publisher between 1869 and 1893; E. J. Mansfield at 190, Strand, was publisher (and at 12, Milford Lane, printer) between 1894 and 1895; G. R. Parker & A. F. Thomas at 190, Strand, were publishers and at 12, Milford Lane printers.<ref name=":6" /> Chief editors were Arthur Locker (1870–1891) and T. H. Joyce (1891–1906).<ref name=":6" /> (Edmund Yates must not have been a chief editor.) In 1890 William Luson Thomas, the same proprietor, spun off a ''Daily Graphic''.<ref>BM [Brian Maidment]. "Thomas, William Luson (1830–1900)." ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland''. Laurel Brake and Marysa Demoor, gen. eds. Gent: Academia Press; London: The British Library, 2009: 623, Col. 2b.</ref> ==== Circulation ==== ''The Graphic'' reported that regular issues in the 1880s occasionally had runs of 250,000, and "Christmas numbers for 1881, 1882 [of] more than 500,000."<ref name=":6" /> ==== ''The Graphic'' Digitized ==== * At the Hathi Trust: https://catalog.hathitrust.org/Record/000533840 * British Newspaper Archive: https://www.britishnewspaperarchive.co.uk/search/results?exactsearch=false&retrievecountrycounts=false&newspapertitle=the%2bgraphic ==== Reading for ''The Graphic'' ==== * Korda, Andrea. ''Printing and Painting the News in Victorian London: The Graphic and Social Realism, 1869–1891''. Ashgate, 2015; Routledge, 2017. ===The Illustrated London News=== The ''Illustrated London News'' was a weekly published on Saturday and costing 6 pence after 1871.<ref name=":6" /> The ''Victorian Fiction Research Guide'' says about the ''Illustrated London News'',<blockquote>by far the most successful of the metropolitan weeklies was a Saturday journal starting up in May 1842, whose most distinctive feature was that it was the first British newspaper to give priority to pictures.<ref name=":4">Law, Graham. "Introduction." ''The Illustrated London News (1842-1901) and The Graphic (1869-1901)''. Retrieved September 2023. https://victorianfictionresearchguides.org/the-illustrated-london-news-and-the-graphic/introduction/.</ref></blockquote>And that by the 1890s it was 32 pages, "small folio size (15.5in x 11.5in), with 3 cols of letterpress," with "over 50 [engravings] from half-column to double-page size.<ref name=":6" /><p> Florence Fenwick-Miller wrote a "Ladies Column," later renamed to "Ladies' Page," for the ''Illustrated London News'':<blockquote>Florence Fenwick-Miller’s weekly ‘Ladies Column’ in ''The Illustrated London News'' and its equivalent in ''The Graphic'', Lady Violet Greville’s ‘Place aux Dames’, form a fascinating contrast. In brief, Fenwick-Miller in ''The Illustrated London News'' takes a progressive line on the suffrage and marriage questions, celebrating a victory for women’s rights in the Jackson/Clitheroe judgement (which denied the authority of the husband to hold his wife against her will, 4 April 1891, 452), yet remains an enthusiastic advocate of the latest feminine fashions from Paris. On the death of Emily Faithful, Fenwick-Miller praises her work as a publisher while criticizing the manliness of her costume (15 June 1895, 750). Greville in ''The Graphic'' opposes electoral or marriage reform, but is in favour of paid work, active athleticism, and rational dress for women – she sees the enfranchisement of women in Australia as the ‘thin end of the wedge’ (25 Nov 1893, 659), but demands that ‘where women do equally good work with men their wages should be the same’ (15 Sept 1894, 306).<ref name=":4" /></blockquote> ==== Proprietors, Publishers, Printers, Editors ==== William J. Ingram & Charles L. N. Ingram were the proprietors between 1872 and 1905 and the publishers and printers between 1884 and 1905.<ref name=":6" /> Chief editors were John Lash Latey (1863-1890), C. K. Shorter (1891-1900) and Bruce S. Ingram (1900-1963).<ref name=":6" /> ==== Circulation ==== The circulation was attested at 123,000 in 1854, with larger runs (as reported by the ''Illustrated London News'') of 310,000 for the issue about the marriage of the Prince of Wales in 1863 (''The Illustrated London News,'' 13 May 1967, 42–3) and of more than 500,000 for holiday issues in the 1880s.<ref name=":6" /> ==== Availability ==== The ILN can be found in Google Books: *Vol. 32, 1858 (https://books.google.com/books?id=FNFCAQAAIAAJ) *Vol. 33, 1858 (https://books.google.com/books?id=ps9CAQAAIAAJ) *Vol. 35, 1859 (https://books.google.com/books?id=3NNCAQAAIAAJ) *Vol. 39, 1861 (https://books.google.com/books?id=V4g-AQAAMAAJ) *Vol. 40, 1862 (https://books.google.com/books?id=yIY-AQAAMAAJ) *Vol. 41, 1862 (https://books.google.com/books?id=xmQjAQAAMAAJ) *Vol. 42, Jan–June 1863 (https://books.google.com/books?id=yoVUAAAAcAAJ or https://books.google.com/books?id=PWUjAQAAMAAJ) *Vol. 45, 1864 (https://books.google.com/books?id=8ok-AQAAMAAJ) *Vol. 46, 1865 (https://books.google.com/books?id=ToY-AQAAMAAJ) *Vol. 47, 1865 (https://books.google.com/books?id=rYk-AQAAMAAJ) *Vol. 89, 1886 (https://books.google.com/books?id=R4o-AQAAMAAJ) *Vol. 91, 1887 (https://books.google.com/books?id=JIo-AQAAMAAJ) *Vol. 92, 1888 (https://books.google.com/books?id=joo-AQAAMAAJ) === The Ladies Field === 1898–1922. The British Newspaper Archive does not have this periodical digitized (as of January 2024). === The Lady === The 1895 ''Mitchell's Newspaper Press Directory'' says ''The Lady'' was composed on a Linotype machine.<ref name=":2" /> (255, Col. 1a) It was published on Wednesdays.<blockquote>LADY. Wednesday. 3''d''. Established February 19, IRRi The ''Lady'' deals with the many subjects in which Iadies are interested fully and completely. Home dress-making, household management, social news, information, hints, and advice, all find place in its pages. It is admirably illustrated with fashions, dresses, &c. Published at 39 & 40, Bedford St., Strand, W.C. (Advt., p. 250.)<ref name=":2" />{{rp|71, Col. 1a}}</blockquote> An advertisement in ''Mitchell's'' for ''The Lady'' says,<blockquote>The Best Ladies' Newspaper. The Lady. Weekly, price Three pence. THE LADY has articles in each issue devoted to the Toilet, the Fashions of Dress, Home Decoration, the Accomplishments, the Social and Domestic Life, Travel for Pleasure and Health, the Household in its many aspects; and numerous other interesting features. A large staff of competent writers, artists, and practical administrators are engaged in each department, with the result that THE LADY is admitted to be best, cheapest, and most useful ladies' journal ever produced. The Terms for Advertisements may be had on application. London — THE LADY Offices, 39–40. Bedford-street and Maiden-lane, Strand. W.C.<ref name=":2" /> (250, Col. 1b)</blockquote>Begun in 1885,<ref>{{Cite journal|date=2023-08-23|title=The Lady (magazine)|url=https://en.wikipedia.org/w/index.php?title=The_Lady_(magazine)&oldid=1171891113|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/The_Lady_(magazine).</ref> the ''Lady'' is still being published, and old issues are not available in a digitized form. The current magazine has a mechanism for getting access to back issues, but they are all 21st-century issues. === The Lady of the House === ''The Lady of the House and Domestic Economist'' began publication on 1 September 1890, the first day of the fall quarter, aimed at educated Irish women and "the Lady Amateur." The first issue says,<blockquote>Introductory. A New Journal which did not claim to fill that time-honoured "long-felt want” which all new Journals seek to occupy would, indeed, show but poor reason for its existence. The Proprietors and Publishers of the “Lady of the House,” although responsible for a new feature in Journalism, have no desire to depart from the traditional custom of the craft. They claim that this Journal distinctly fills a long-felt want, and fills it well. The want has long been felt of a high-class Irish Journal solely devoted to Fashion, the Beautifying of the Home and Person, Scientific Cookery, the Toilet, the Wants and Amusements of Children, the Garden and Conservatory, and the hundred-and-one matters which interest educated women. This want, we repeat, has been felt, but has not hitherto been filled, except by the English Ladies’ Journals, which enjoy an immense circulation in this country. The “Lady of the House” will be issued Quarterly — on the first day of each season — Autumn, Winter, Spring, and Summer. The Autumn Number is now presented, and comprises Fashions for Autumn, Seasonable Descriptions of New Hats, Gowns, Mantles, &. Dishes for Autumn will be found in the "Cookery Section;" a high Art Authority describes the best arrangement of the house in Autumn, and a no less high Authority on Horticulture instructs the Lady Amateur on the management of her Garden at this Season. This, the plan on which the Journal is originated, will be fully and faithfully observed each Quarter, when ''Twenty Thousand Copies'' will be distributed gratuitously. The costliness of such an undertaking must be apparent to everyone. Notwithstanding this, the Proprietors do not seek the Subscriptions of the reading public. The next (Winter Quarter) Number will be issued on the first day of Winter — 23rd December next — and will contain an exhaustive ''résumé'' of the Paris Winter Fashions, and a mass of finely-illustrated Literature, suitable for Christmastide.<ref>"Introductory." ''Lady of the House'' 1 September 1890, Monday: 3 [of 38], Col. 1a–2c [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0004835/18900901/012/0003''.''</ref></blockquote> === Lloyd's Weekly Newspaper === 1842–1931. Edited by Thomas Catling 1884–1906.<ref name=":7">{{Cite journal|date=2023-09-02|title=Lloyd's Weekly Newspaper|url=https://en.wikipedia.org/w/index.php?title=Lloyd%27s_Weekly_Newspaper&oldid=1173436602|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lloyd%27s_Weekly_Newspaper.</ref> "On 16 February 1896, ''Lloyd’s Weekly'' became the only British newspaper in the nineteenth century to sell more than a million copies."<ref name=":7" /> === The London Gazette === An official journal of record for the government of the U.K., the London Gazette has detailed coverage of official social events — like weddings of the royal family, for example, and granting of awards and honors. * Front page: https://www.thegazette.co.uk. * Number 23720, 24 March 1871, is a supplement detailing the wedding of [[Social Victorians/People/Princess Louise|Princess Louise]] and John Campbell, Marquis of Lorne (https://www.thegazette.co.uk/London/issue/23720/) * Number 26869, 2 July 1897, records nothing about the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] because nothing official occurred because of it (https://www.thegazette.co.uk/London/issue/26869/page/3637). === The Man of Ross === Also, ''The Man of Ross, Forest of Dean, and West of England Advertiser''. A conservative newspaper that came out on Saturday, 1d. (one penny).<ref name=":2" />{{rp|145, Col. 1b}} The 1895 Mitchell's says of Ross, Herefordshire,<blockquote>A market town, with iron and coal-mines in the neighbourhood, and extensive iron and tinplate-works about six miles distant. The district is rural and the population (9,651) is engaged in mining and agricultural pursuits.<ref name=":2" /> (145, Col. 1b)</blockquote>Of the Man of Ross newspaper, Mitchell's says,<blockquote>Gives the local and general news of the week, with a varied, useful, and entertaining miscellany of general information, and original articles. P<small>ROPRIETOR</small> — John Counsell.<ref name=":2" />{{rp|145, Col. 1b}}</blockquote> ===The (London) Morning Post=== In 1879, Mitchell's Press Directory described the Morning Post as follows: <blockquote>MORNING POST. Daily, 3d. Established 1772. Principles: High Church and Whig. The Post is not merely a political newspaper, it is the fashionable chronicle and journal of the Beau Monde. Few events occur in the higher circles, to which publicity can consistently be given, which are not reported in its columns. Its news department is full and complete; its reports impartial, and well written; and its criticisms on books, music, pictures, and science are considered as authorities. Its correspondents are numerous; and those in the colonies especially are evidently well informed upon all questions that form the subjects of public discussion of government policy. It is an able and consistent advocate of the principles of the "High Church" party, as distinguished from the "Evangelical" section of the Church; but it does not favour the doctrines of the Ritualistic party. Published by F. W. Smith, Wellington Street, W.C. (Gliserman [11])</blockquote> Brake and Demoor say the ''Morning Post'' was taken over by Peter Borthwick in 1849 and bought by his son Algernon Borthwick, who had been editor as well, in 1879. In 1895 ''Mitchell's Newspaper Press Directory'' says that the ''Morning Post'''s politics were conservative, the "Latest Time for Ads." was 10 a.m., and the "Time Published" was 3 p.m.<ref name=":2" />{{rp|55}}. Laurel Brake and Marysa Demoor's ''Dictionary of Nineteenth-Century Journalism in Great Britain and Ireland'' says the following: <blockquote>The editorship was taken over by Peter Borthwick in 1849, the start of a family connection that was to last until 1924. On Borthwick's death in 1852, the editorship passed to his son Algernon Borthwick, who bought the paper in 1876, and consolidated its imperialist* and conservative tone. He also continued its interest in sporting* matters, particular racing. When he took over the paper, its circulation had declined to under 3,000 (compared to a circulation of The Times of 40,000.) He reduced the price* from 3d to 1d and increased its circulation. During his editorship, leader writers included Andrew Lang* and Alfred Austin*. William E. Henley*, Thomas Hardy* and Rudyard Kipling contributed verse while George Meredith was its special correspondent during the Italian wars* of liberation from Austria. Borthwick, now Lord Glenesk, died in 1908 and his family sold the paper in 1924. It merged* with the Daily Telegraph* in 1937. JRW Sources: Griffiths 1992, Hindle 1937, ODNB. (Brake and Demoor 427; asterisks sic, references to articles in the book)</blockquote> In Mitchell's 1906 ''Newspaper Press Directory'', the ''Morning Post'' is described as follows: <blockquote>Morning Post. Daily, 1d. Established 1772. Principles: Unionist. The Morning Post is the oldest daily newspaper existing in London. It gives special attention to fashionable and foreign news, and is also noted for its full and accurate reports of Parliamentary proceedings. As a medium for announcements which it is desired to bring before the notice of the high and wealthy classes, the Morning Post cannot be surpassed. Published by E. E. Peacock, Aldwych, W.D. (Advt. p. 88.)<br> Tele. Nos.<br> Strand (5432 Gerrard.<br> (13553 P.O. Central<br> Aldwych, 13501 P.O. Central<br> City Office, 5522 Avenue. (NPD 1905: 62; identical description in Mitchell 1896 58)</blockquote> ''Willing's British and Irish Press Guide'' for 1891 describes the Morning Post like this: <blockquote>MORNING POST, 1772. (c) Daily — 3 a.m. 1d. T. L. Coward, 12 Wellington Street, W.C. Political, general, and fashionable newspaper. (Willing's 1891: 79)</blockquote> Willing's also classifies the ''Morning Post'' as a family newspaper.{{rp|135}} <blockquote>Sell's Dictionary of the World's Press says this in 1886: Dating its birth back to the year 1772, this paper can boast of being the oldest political daily newspaper existing in London. Its career has ben a very distinguished and interesting one; and among its contributors it has numbered Southey, Wordsworth, Sir James Mackintosh and others. Coleridge was for some time its editor, and Charles Lamb contributed witty paragraphs. From its commencement it has been most ably conducted, and its criticisms on plays, music, and books are excellent. The special features of the Morning Post are its fashionable and foreign news, to which it gives special [127/128] prominence. Nothing of interest occurs in the upper circles of society that is not recorded in its columns, and everything which can interest the beau monde receives notice. The circulation of the Morning Post, though not so great as some of its contemporaries, is a very good one, being chiefly among fashionable and wealthy circles. This paper is consequently well adapted for the advertising of articles de luxe and good possessing first-class workmanship and artistic merit, the sale of which is almost entirely confined to persons to whom the cost is of secondary importance. Compared with the other "dailies" the advertising charges of the Morning Post are moderate. Till within the last two years this paper was published at threepence, but now its price is the general one of a penny, a reduction which has already increased its sale tenfold.{{rp|127–128}}</blockquote> Advertising prices for the Morning Post from the Newspaper Press Dictionary (NPD 1905: 88), found in Google Books: [IMG] ====The ''Morning Post'' in Fiction==== When Major Pendennis moves to the country in Thackeray's 1864 novel, "he will miss seeing his name in the Morning Post on the day after each of the 'great London entertainments'" (Hampton, Mark. Visions of the Press in Britain, 1850–1950. Urbana and Chicago, IL: University of Illinois Press, 2004: 23). Gwendolyn in Wilde's ''The Importance of Being Earnest'' says she will announce her engagement in the ''Morning Post''. In a discussion of parodies of newspaper journalism, Patrick Leary says, "Punch frequently ran such parodies, beginning quite early on in the 1840s. The obsequiousness of the Morning Post (or "The Fawning Post," as Douglas Jerrold liked to call it) was a favorite target." (Leary). ====Some Important Writers, Contributors, Editors, Etc.==== *C. J. (Charles James) Dunphie was art and theatre critic 1856–1908 (Brake and Demoor 186) *William A. Barrett was "chief music* critic on the Morning Post* (1866–1891)" (Brake and Demoor 39) *Algernon Borthwick founded a "society magazine" called The Owl (Brake and Demoor 67) *Florence Caroline Douglas Dixie, war correspondent in the Boer War, 1897 (Brake and Demoor 172) *Rudyard Kipling *Benjamin Disraeli, before Borthwick took over (Brake and Demoor 427) *Andrew Lang, occasional contributor (Brake and Demoor 346) *Alfred Charles William Harmsworth, Viscount Northcliffe (Brake and Demoor 270) *William E. Henley (Brake and Demoor 427) *Alfred Austin (Brake and Demoor 427) *Thomas Hardy (Brake and Demoor 427) *George Meredith (Brake and Demoor 427) *Winston Churchill (Brake and Demoor 412) ===The Pall Mall Gazette=== Mitchell's classifies the ''Pall Mall Gazette'' as an evening paper.<ref name=":2" /> (p. 60, Col. 1b) The ''Pall Mall Gazette'' ran a "ladies' column" called the "Wares of Autolycus"<blockquote>from May 1893 to the end of 1898, appearing most days of the week, and drawing on a group of female journalists, notably Alice Maynell, to cover between them literature, gardening, fashion, home decor, good food, and society news. But though constructed in gossip column form, its aesthetic and literary standards lifted it well above the level of the average contemporary gossip column.<ref name=":5">Onslow, Barbara. "The Ladies' Page." ''Victorian Page: The Web Magazine of Victoriana'' http://www.victorianpage.com/VictorianPage-Ladiespage-womensmagazines.html (accessed April 2017).</ref></blockquote>Both [[Social Victorians/People/George Bernard Shaw|George Bernard Shaw]] and [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] wrote for the ''Pall Mall Gazette'', which was edited by W. T. Stead. Shaw wrote book reviews. Special issues of the ''Pall Mall Gazette'' published some investigative journalism Stead did, "The Maiden Tribute to the Modern Babylon," about selling girls for sexual slavery (which lead to the Criminal Law Amendment Act of 1885). ===The Pictorial World=== The Pictorial World was an illustrated weekly newspaper that published between 7 March 1874 and 9 July 1892, or perhaps a new series began in 1891 (conflicting library records). According to its first issue, <blockquote> The Programme of The Pictorial World may be given in a few words. It is to present to the great middle-class of England, and of all English-speaking countries, a weekly illustrated record of passing events, which shall be pure in tone, amusing in its contents, and graceful to the eye— a paper which will depict faithfully with pen and pencil both "what the world says" and "what the world does." In The Pictorial World authors and artists will work together— each will inspire the other; and the cut-and-dried style of article shall be as much as possible avoided. It will therefore largely depend upon external help and kindnesses, and will open its pages to interesting sketches, far-brought novelties, and hints from friends at home and abroad. Such, in brief outline, is our wish and plan: we offer this first number as an earnest of our desire to carry it out; our succeeding numbers will show a progressive improvement. Appealing for public support, we look confidently to the future. (1884-03-07 Pictorial World) </blockquote> Lady Violet Greville says she wrote anonymously or pseudonymously for the ''Pictorial World'' (1894-04-04 Sketch 5, Col. 1C), perhaps shortly after it began publication. Mary Elizabeth Braddon published ''The Golden Calf'' in the ''Pictorial World'', 1882–1883. George Robert Sims published a series called "How the Poor Live" beginning in 1883. ===The Queen, The Lady's Newspaper=== The weekly newspaper (published on Saturdays) ''Queen'' was marketed to women in the "upper ten thousand," an expression originally used for American Society but later translated to the U.K. Through a couple of major changes, the last major one of which occurred in 1970, what was the ''Queen'' is now ''Harper's Bazaar''. A column called "The Upper Ten Thousand at Home and Abroad" appeared regularly by the end of the 19th century detailing the movements and social events of the royals, aristocracy, political leaders and plutocrats. Ardern Holt seems to have been the major writer for fashion, at least in 1897, including an advice column for fashion, dress and costumes. Mitchell's ''Newspaper Press Directory, 1895'' has this for its entry for the ''Queen'': <blockquote>Q<small>UEEN</small>. Saturday, 6''d''. Established 1861. P<small>RINCIPLES</small>: Neutral. It is particularly intended for ladies' reading, as it provides that which ladies have hitherto so much needed in this country; the ''earliest'' colored fashion-plates from Paris, and original work-patterns by the best designers. It has many novel departments, in which ladies communicate useful observations and criticisms. "Pastimes," "Domestic and Rural Economy," and "Domestic Pets," are also included; and a large space is given to "Receipts" for family use. Pastimes for ladies, a charade, a novel, or a sprightly sketch, vary the contents. Court and fashionable news are fully reported and the paper is well illustrated. Published by Horace Cox, Bream's Buildings, Chancery Lane, E.C. (Advt., p. 252.)<ref name=":2" />{{rp|75}}</blockquote>The ad for the ''Queen'' in Mitchell's ''Newspaper Press Directory, 1895'' looks like this: <blockquote>Queen, the Lady's Newspaper. This newspaper is the great organ of the ladies of the upper classes in Great Britain. The latest Paris and other fashions are given every week, together with patterns and descriptions of the newest work, illustrated in the best style of art. The following list will show the chief features of the paper: — [what follows is a 2-column list with a vertical rule between the 2 columns, which break after "Society in Paris" and before "Work of all kinds."] :Leaders on interesting and current topics :The Exchange :Dramatic critiques :Paris and other fashions :Gleanings from new books :The boudoir :The housekeeper :The opera, concerts, &c. :Society in Paris :Work of all kinds :Plants and flowers :Recipes of all kind :New music :Natural History :Court news :Pastimes :New books :Literary and artistic gossip. The QUEEN is also the great medium through which tradesmen and others bring their announcements prominently before the upper ten thousand. These advertisements comprise, among other subjects — dress and fashion, country wants, governesses, schools, books, furniture, pastimes, domestic wants, music, toilet requisites, servants, &c. In addition to the above, the QUEEN presents a monthly coloured fashion sheet and monthly coloured work patterns, a monthly cut paper pattern, and illustrations (coloured and plain) of all new fancy work, domestic inventions, fashions, &c. Prince 6c.; stamped 6 1/2d.; yearly subscription, pain in advance, 28s.; half yearly, 14s.; quarterly, 7s. Specimen copy post free for six stamps. Published every Saturday by Horace Cox, Bream's-buildings, Chancery-lane.<ref name=":2" />{{rp|252, Col. 3a}}</blockquote> === Society === In a posting on the Victoria listserv, Patrick Leary says,<blockquote>According to the ''Waterloo Directory'', the penny weekly magazine ''Society'' ran from 1878 to 1890.  The editor was George Plant, and it was printed by Unwin Brothers.  The entry lists [illustrator] Phil May as a contributor. I couldn't find the journal online — that generic title is hard to zero in on — but the British Library has a full run of the paper; the Bodleian has a partial one.   <p> Fox-Bourne's history of the press has a little bit more about ''Society'' here https://babel.hathitrust.org/cgi/pt?id=uga1.32108003235689&seq=325.<ref>Leary, Patrick. "Re: [VICTORIA] Phil May." ''Victoria: The Online Discussion Forum for Victorian Studies.'' 14 July 2024.</ref></blockquote>In a reply to the same thread on the Victoria listserv, Richard Fulton says,<blockquote>The ''Union List of Victorian Serials'' lists ''Society'' as running under that title from 12 mar 1880 to 31 Aug 1901. It also notes that the magazine started out life in 1879 as the ''Mail Budget''.<ref>Fulton, Richard. "Re: [VICTORIA] Phil May." ''Victoria: The Online Discussion Forum for Victorian Studies.'' 15 July 2024.</ref></blockquote> === The (London) Standard and Evening Standard === The London ''Standard'' was the first of these two newspapers, founded in 1827.<ref name=":3">JRW [John Richard Wood]. "''Standard'' (1827–1916)." ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland''. Laurel Brake and Marysa Demoor, gen. eds. Gent: Academia Press; London: The British Library, 2009: 596, Col. 2c – 597, Col. 1a.</ref>{{rp|596, Col. 2c}} An advertisement in Mitchell's for "The Standard, Morning and Evening," says that it is "the leading daily newspaper" and<blockquote>contains full Parliamentary, Law, Police, and Commercial Intelligence, together with Critiques on all noteworthy productions in the worlds of Art, Literature, Music, and the Drama, and a carefully-revised Epitome of the general News of the day.<ref name=":2" />{{rp|81}}</blockquote> Addresses: 103, 104 and 105 Shoe Lane and 23 Bride Street, London, E.C. ==== London Standard ==== The London ''Standard'' became a daily paper in 1857. In the 19th century, the ''Standard'' and the ''Morning Standard'' are the same paper. ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland'' says of the London ''Standard'',<blockquote>in 1878 the paper passed into the control and editorship of William Heseltine Mudford and by the mid-1880's the / Standard had become a powerful force in conservative journalism* with a circulation of 250,000. Its leader* writers included Alfred Austin* and Thomas Escott*. George Alfred Henty, the author of stories for boys, was its war* correspondent*.<ref name=":3" />{{rp|596, Col. 2c – 597, Col. 1a; asterisks sic, references to articles in the book}}</blockquote> The 1895 ''Mitchell''<nowiki/>'s says,<blockquote>S<small>TANDARD</small>. Daily, 1''d''. Established as a Morning Paper, June 29, 1857. P<small>RINCIPLES</small>: Conservative. While maintaining Conservative principles, ''The Standard'' reserves the right to apply those principles to the questions of the day, without regard to party politics, or special devotion to the views of party leaders. On all political questions it is remarkably impartial in the admission to its columns of letters from any man whose position gives him a right to speak, be his views what they may. In the matter of Parliamentory news ''The Standard'' is the one London Penny Journal that has not adopted the system of very abridged reports. The paper has of late paid great attention to foreign correspondence: more particularly such as is forwarded by telegraph from all parts of the world. In literary and dramatic criticism it exercises a careful selection of productions worthy of notice for praise or blame; but the complete display of '''him''' and foreign news is its chief distinguishing feature. Reports relating to markets, racing, cricket, and boating are very fully given. Published by A. Gibbs, 104, Shoe Lane, E.C. (Advt., p. 81)<ref name=":2">Mitchell, Charles. ''Newspaper Press Directory, 1895''. [Hathi Trust via U Wisconsin Madison.] London: C. Mitchell & Co., 1895. http://hdl.handle.net/2027/mdp.39015085486150 (accessed January 2023).</ref>{{rp|60, Col. 1b}}</blockquote> ==== London Evening Standard ==== ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland'' says, "The ''Evening Standard'' was issued as a sister newspaper [of the London ''Standard''] in 1860."<ref name=":3" />{{rp|596, Col. 2c}}. From Brake and Demoor: The ''Pall Mall Gazette'': "only to be dissolved in 1923 into Lord Beaverbrook's ''Evening Standard''" (478, Col. 1c). The ''Standard'': "the paper was acquired by C. Arthur Pearson* in 1904, when its circulation was 80,000. The ''Standard'' ceased publication in 1916, but the ''Evening Standard'' continued"{{rp|597, Col. 1a}}. The 1895 Mitchell's says,<blockquote>E<small>VENING STANDARD</small>. Daily, 1''d'', Estab. 1827. P<small>RINCIPLES</small>: Conservative. Under the same management as the Standard published in the morning. Published by A. Gibbs, 104, Shoe Lane, E.C. (Advt., p. 81.)<ref name=":2" /> (60, Col. 1c)</blockquote>An advertisement for the ''Evening Standard'' says that although it was an evening paper, it published 4 editions, the last (or "Latest") must have been very late:<blockquote>Published four times daily, gives the Day's Law, '''Police''', Markets, Commercial Meetings, Stock Exchange Quotations, &c. The Latest or "S<small>PECIAL</small>" Edition contains, in addition, the Day's Racing, and (during the Parliamentary Session) a full Summary of the Debates in both Houses of Parliament.<ref name=":2" />{{rp|81}}</blockquote> === The Star of Guernsey === Not to be confused with the radical paper ''The Star'', the ''Star of Guernsey'', as the 1895 Mitchell's says,<blockquote>Is published every Tuesday, Thursday, and Saturday, price 1d., or by post 1¹/₂d. to any part of the United Kingdom, France, and most parts of the Continent.<p> The STAR circulates very extensively through the Channel Islands, and large numbers are sent to the United Kingdom, the Colonies, France and America, it is, therefore, an excellent medium for advertisers.<ref name=":2" /> (315, Col. 3a)</blockquote>The proprietors were Marquand & Co. STAR Office, Guernsey. === The St. James's Gazette === The 1895 ''Mitchell's Newspaper Press Directory'' says the ''St. James's Gazette'' was published at 3:00 p.m.<ref name=":2" />{{rp|56a}}<blockquote>S<small>T. JAMES'S GAZETTE</small>. 1''d''. Established 1880. The ''St. James's Gazette'' is an independent and progressive Conservative newspaper, which, while consistently supporting constitutional principles, the maintenance of the empire, and the supremacy of the law in every portion of the dominions of the Crown, is in favour of moderate and ordered reform. It gives with point, brevity, and accuracy all the most important news of the day, the latest money market reports, racing news, Parliamentary Intelligence, Police News, Foreign Telegrams, &c. Special attention is given to American, Continental, and Indian Intelligence. Published at Dorset Street, Whitefriars.<ref name=":2" />{{rp|60, Col. 2b}}</blockquote> ===Sussex Agricultural Express=== The ''Sussex Agricultural Express'', in describing a social event in which the Duke and Duchess of Devonshire, as Mayor and Mayoress, decorated Devonshire House again, refers to some of the men who worked for the Duke and Duchess in January 1898: "Mr. J. P. Cockerell, the Duke of Devonshire's indefatigable agent called to his aid a willing and competent staff from Compton Place, including Mr. W. S. Lawrence, the house steward, and Mr. May, the gardener."<ref>"Sunday School Festival: Speech by the Duke." ''Sussex Agricultural Express'' 29 January 1898, Saturday: 7 [of 12], Col. 5b–6a. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000654/18980129/182/0007.</ref> === Vanity Fair === Not the American magazine, a society magazine (7 November 1868 – 5 February 1914).<ref>{{Cite journal|date=2023-12-26|title=Vanity Fair (British magazine)|url=https://en.wikipedia.org/w/index.php?title=Vanity_Fair_(British_magazine)&oldid=1191870176|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Vanity_Fair_(British_magazine).</ref> The caricature portraits<ref>{{Cite journal|date=2022-06-01|title=List of Vanity Fair (British magazine) caricatures|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures&oldid=1090963973|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures.</ref> of famous men and, occasionally, women were and continue to be an important contribution by this magazine, by people like Spy and Drawl (Leslie Ward) and Max Beerbohm, including other notable artists. ===The Times=== The 1895 ''Mitchell's Newspaper Press Directory'' includes the ''Times'' among the morning papers: <blockquote>The Times. Daily, 3d. Established January 1, 1788, (weekly edition, 2d., established January, 1877.) Principles: Church of England in religion; Free Trade in mercantile and commercial transactions. This, the leading journal of Europe, has for the field of its circulation, emphatically, the WORLD, and its influence is co-extensive with civilization. The connection is clear between the circulation and the advertisements. Not so clear is the relation between the circulation and the influence: to some extent the influence may be the effect; but chiefly, we suspect, the cause. The consciousness that thousands upon thousands read, creates some impression, an idea which may be to some extent the source of influence and of power. But there is in the influence of the Times something more substantial, more potent, than can be accounted for by the mere consciousness of its enormous circulation; it is "looked up to" all over Europe, and it is the only paper which men of all parties, and all classes, read and speak of. Other papers may be more preferred by particular classes, but all read the Times, who can; just because it is not possible to predicate its course on any question as regulated by the interest of any party or class: and it is known that it always acts on views of its own. It deals out its denunciations with equal force and freedom on all parties in their turn, with a boldness and decision quite characteristic; but not unfrequently, with great indifference to the consistency of its opinions. Hence all parties are uncertain what next they may exult in, a fiery storm invective against their antagonists or suffer the infliction themselves. It is distinguished for its reports of parliamentary and legal proceedings. It does not devote much of its space to literature and the fine arts; but its reviews and criticisms are forcibly and cleverly written. Published by G. E. Wright Printing House Square, E.C.<ref name=":2" />{{rp|57}}</blockquote> Costing 3d. per daily issue, the "Latest Time for Ads." for the ''Times'' was 7 p.m., and the "Time Published" was 5 a.m.<ref name=":2" />{{rp|56}} === Westminster Gazette === George Newnes founded the ''Westminster Gazette'' in 1893 as the "radical liberal successor" to the ''Pall Mall Gazette'', after it had been purchased "by Tory interests."<ref name=":1">JRW [John Richard Wood]. "Westminster Gazette." ''The Dictionary of Nineteenth Century Journalism in Great Britain and Northern Ireland''. Laurel Brake and Marysa Demoor, gen. eds. Gent: Academia Press; London: The British Library, 2009: 672, Col. 1c.</ref> A "'clubland' 1d evening daily," it was called the "pea-green incorruptible" (the pea-green because of the green paper it was printed on).<ref name=":1" /> The ''Westminster Gazette'' merged with the ''Daily News'' in 1928.<ref name=":1" /> Edward Tays Cook was editor 1893–1895, and John Alfred Spender 1895–1928.<ref name=":1" /> ===The Woman's World=== November 1887 – Editor, Oscar Wilde (April 1887 – by October 1889) Sub-Editor, Arthur Fish ''The Woman's World'' ceased publication not long after Wilde left it. ''The Queen'' was a competitor. [[Social Victorians/People/Oscar Wilde|Oscar Wilde]] took over the editorship of ''The Lady's World'' in April 1887, changing its title (to ''The Woman's World'') and its mission.<ref name=":8">Fitzsimons, Eleanor. "Entering ''The Woman’s World'': Oscar Wilde as Editor of a Woman’s Magazine." ''The Victorian Web'' 17 September 2015. http://www.victorianweb.org/. Rpt. ''Academia'' https://www.academia.edu/15813341/Entering_The_Woman_s_World_Oscar_Wilde_as_Editor_of_a_Woman_s_Magazine. Rpt.? Eleanor Fitzsimons, ''Wilde's Women: How Oscar Wilde Was Shaped by the Women He Knew'' (Duckworth Overlook, 2015).</ref> ''The Lady's World'' was "a high-end, illustrated monthly magazine produced by Cassell and Company" that focused on fashion.<ref name=":8" /> ''The Woman's World'' was significantly redesigned for its November 1887 first issue:<blockquote>A fresh cover design featured Wilde’s name prominently with key contributors listed below. In a significant departure from convention, each article was attributed to its author by name. Wilde also increased the page count from thirty-six to forty-eight, and relegated fashion to the back while promoting literature, art, travel and social studies. Gone entirely were ‘Fashionable Marriages’, ‘Society Pleasures’, ‘Pastimes for Ladies’ and ‘Five o’clock Tea’. In his ‘Literary and Other Notes’, Wilde demonstrated unequivocal support for the greater participation of women in public life.<ref name=":8" /></blockquote>The "keynote" of ''The Woman's World'', according to Arthur Fish, was "the right of woman to equality of treatment with man."<ref name=":8" /> Wilde wrote to Thomas Wemyss Reid, General Manager of Cassells, that he wanted ''The Woman's World'' to "take a wider range, as well as a high standpoint, and deal not merely with what women wear, but with what they think, and what they feel."{{rp|qtd. in}}<ref name=":8" /> Eleanor Fitzsimons looks at the new way the periodical treated women's fashion under Wilde's editorship: "Although fashion remained a key feature, a conventional round-up of the season’s trends was supplemented with articles on cross-dressing, aesthetic design and rational dress."<ref name=":8" /> ===The World: A Journal for Men and Women=== The first number of the ''World'' was 8 July 1874. Edmund Yates and E. C. Grenville Murray were proprietors until 1874; Yates was editor from the beginning until the end of his life in 1894.<ref>Edwards, P. D. "Journalism." "Edmund Yates." ''Victorian Fiction Research Guides''. No. 3. Online. Accessed April 2017: http://victorianfictionresearchguides.org/edmund-yates/journalism/.</ref> Yates wrote editorials under the pseudonym Atlas. According to P. D. Edwards, the ''World'' was <blockquote>a weekly newspaper dedicated to the style of ‘personal journalism’ that Yates had been perfecting in his various gossip columns for nearly twenty years. Its appeal was to men and women of the world: clubmen, sportsmen, hangers-on of the literary, theatrical, and artistic worlds, fashionable and would-be fashionable ladies. After a few months it became a conspicuous and continuing success, generating hosts of imitators and inaugurating, it is generally agreed, the most distinctive twentieth-century style of journalism.<ref>Edwards, P. D. "Introduction." "Edmund Yates." ''Victorian Fiction Research Guides''. No. 3. Online. Accessed April 2017: http://victorianfictionresearchguides.org/edmund-yates/.</ref></blockquote> Some of the people who wrote for the ''World'' during Yates' editorship were [[Social Victorians/People/George Bernard Shaw|G. B. Shaw]], [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], and so on. It looks like the ''Clifton Society'' reprinted "What the World Says" columns from ''The World''. == Earlier in the Century == === The Court Journal, and Gazette of the Fashionable World, Literature, Music and the Fine Arts === Google Books has a few volumes of this paper. It was a weekly, 3 columns, 6''d''. per issue, 6''s''. 6''d''. per quarter. Publishing Office: 21, Catherine-street, The Strand. Ads at the end of each issue, ~15 pages. It had a section called "Court and Fashionable Gossip." There's no ''Wikipedia'' page on it, so I'm not certain of the run, but the issue dated 2 April 1853 is No. 1243, No. 264 New Series. Google Books has # 1833 (https://books.google.com/books?id=2KYEo3j3YL8C) # 1835 (https://books.google.com/books?id=LLcRAAAAYAAJ) # 1848 (https://books.google.com/books?id=4pIechTAkPIC) # 1853 (https://books.google.com/books?id=JKhUGEnNVTwC) # 1854 (https://books.google.com/books?id=naw0BY8lYh8C) # 1858 (https://books.google.com/books?id=JhJ_hI-lxCsC) # 1859 (https://books.google.com/books?id=1VcG8C2nbv4C) The 1853 volume has 40 issues. # ''The Court Journal, and Gazette of the Fashionable World, Literature, Music and the Fine Arts'', 2 April 1853 (No. 1243; No. 263 New Series): . https://books.google.com/books?id=JKhUGEnNVTwC # ''The Court Journal ...'', 9 April 1853 (No. 1244; No. 264 New Series): 225–240. # ''The Court Journal ...'', 16 April 1853 (No. 1245; No. 265 New Series): 241–256. # ''The Court Journal ...'', 23 April 1853 (No. 1246; No. 266 New Series): 257–272. # ''The Court Journal ...'', 1 May 1853 (No. 1247; No. 267 New Series): 273–288. # ''The Court Journal ...'', 7 May 1853 (No. 1248; No. 268 New Series): 289–304. # ''The Court Journal ...'', 14 May 1853 (No. 1249; No. 269 New Series): 305–320. # ''The Court Journal ...'', 21 May 1853 (No. 1250; No. 270 New Series): 321–336. # ''The Court Journal ...'', 28 May 1853 (No. 1251; No. 271 New Series): 337–352. # ''The Court Journal ...'', 4 June 1853 (No. 1252; No. 272 New Series): 353–376. # ''The Court Journal ...'', 11 June 1853 (No. 1253; No. 273 New Series): 377–392. # ''The Court Journal ...'', 18 June 1853 (No. 1254; No. 274 New Series): 393–416. # ''The Court Journal ...'', 25 June 1854 (No. 1255; No. 275 New Series): 415–440. # ''The Court Journal ...'', 2 July 1854 (No. 1256; No. 276 New Series): 441–456. # ''The Court Journal ...'', 9 July 1854 (No. 1257; No. 277 New Series): 457–472. # ''The Court Journal ...'', 16 July 1854 (No. 1258; No. 278 New Series): 473–488. # ''The Court Journal ...'', 23 July 1854 (No. 1259; No. 279 New Series): 489–504. # ''The Court Journal ...'', 30 July 1854 (No. 1260; No. 280 New Series): 505–520. # ''The Court Journal ...'', 6 August 1854 (No. 1261; No. 281 New Series): 521–536. # ''The Court Journal ...'', 13 August 1854 (No. 1262; No. 282 New Series): 537–552. # ''The Court Journal ...'', 20 August 1854 (No. 1263; No. 283 New Series): 553–568. # ''The Court Journal ...'', 27 August 1854 (No. 1264; No. 284 New Series): 569–584. # ''The Court Journal ...'', 3 September 1854 (No. 1265; No. 285 New Series): 585–600. # ''The Court Journal ...'', 10 September 1854 (No. 1266; No. 286 New Series): 601–616. # ''The Court Journal ...'', 17 September 1854 (No. 1267; No. 287 New Series): 617–632. # ''The Court Journal ...'', 24 September 1854 (No. 1268; No. 288 New Series): 633–648. # ''The Court Journal ...'', 1 October 1854 (No. 1269; No. 289 New Series): 649–664. # ''The Court Journal ...'', 8 October 1854 (No. 1270; No. 290 New Series): 665–680. # ''The Court Journal ...'', 15 October 1854 (No. 1271; No. 291 New Series): 681–696. # ''The Court Journal ...'', 22 October 1854 (No. 1272; No. 292 New Series): 697–712. # ''The Court Journal ...'', 29 October 1854 (No. 1273; No. 293 New Series): 713–728. # ''The Court Journal ...'', 5 November 1854 (No. 1274; No. 294 New Series): 729–744. # ''The Court Journal ...'', 12 November '''1853''' (No. 1275; No. 295 New Series): 745–760. # ''The Court Journal ...'', 19 November 1853 (No. 1276; No. 296 New Series): 761–776. # ''The Court Journal ...'', 26 November 1853 (No. 1277; No. 297 New Series): 777–792. # ''The Court Journal ...'', 3 December 1853 (No. 1278; No. 298 New Series): 793–808. # ''The Court Journal ...'', 10 December 1853 (No. 1279; No. 299 New Series): 809–824. # ''The Court Journal ...'', 17 December 1853 (No. 1280; No. 300 New Series): 825–840. # ''The Court Journal ...'', 24 December 1853 (No. 1281; No. 301 New Series): 841–856. # ''The Court Journal ...'', 31 December 1853 (No. 1282; No. 302 New Series): 857–872. == Magazines and Other Periodicals == === The Lady === Founded by Thomas Gibson Bowles (1842–1922) in 1885, The Lady is still in publication. I haven't found any copies of 19th-century issues. === The Lady's Realm: An Illustrated Monthly Magazine === Gossipy, with a focus on the aristocracy and fashionable and news about the Season. Some fiction and poetry, mostly written by women with titles. * 1898, May–October, Vol. IV (): https://books.google.com/books?id=KG8-AAAAYAAJ * 1899, May–October, Vol. VI (): https://books.google.com/books?id=LG4-AAAAYAAJ * 1900, November–April, Vol. IX (): * 1901–1902, November–April, Vol. XI (): https://books.google.com/books?id=94x2MboTkX8C === London Society: A Monthly Magazine of Light and Amusing Literature for the Hours of Relaxation === A lot of serialized fiction, but Alexander Henry Wylie seems to have had an article in each issue about Society in one way or another. * 1889, July–December, Vol. LVI (56): https://books.google.com/books?id=oz0ZAAAAYAAJ * 1890, January–June, Vol. LVII (57): https://books.google.com/books?id=tSZKAAAAMAAJ * 1890 July–December, Vol. LVIII (58): https://books.google.com/books?id=-zIZAAAAYAAJ * * 1892 July–December, Vol. LXII (62): https://books.google.com/books?id=A1GAbrVRCDUC * etc. ==== Alexander Henry Wylie, "Society in 1892." ''London Society'' December 1892 (Vol. LXII): 611–614. ==== Anti-Semitism alert; classism alert. <blockquote> SO much has been written by Lady Cork, Lady Jeune, Mr. Mallock, and other writers on "society," that it seems superfluous to add anything to what they have contributed to various magazines; but to an on-looker who does not go to "every lighted candle " the question naturally arises, What is now called "society?" There was a time, say, thirty years ago, when undoubtedly there was such a thing, leaving out, of course, the political ladies, who owed it to their party and their husbands to entertain ''all'' that were "on their side of the House." That we leave entirely alone, although in the case of Lady Palmerston (who stands alone, as a political lady, from an entertaining point of view), she steered clear of receiving any one who was not a friend, a relation, a person of birth and position, a ''great'' luminary in the political world, a celebrated author, or in some way ''entitled'' to an invitation to the best ''salon'' the London world has seen for many generations, and, so far, is ever likely to see again. Frances, Lady Waldegrave had a ''salon'', but of a totally different kind: pleasant, yes, certainly; but cosmopolitan, undoubtedly. A loss she certainly is, not to the "great world," but to those who in every sense almost were her inferiors, and who would like to go out every night of their lives in a frivolous round of what they call "society." But I maintain "society " of thirty years ago does not exist at the present day. One most important cause is, notwithstanding what may be said to the contrary — and there are those who must own it to themselves — "You forget we have daughters to marry." No, I do not forget it, but strongly maintain all the more, considering the ''present'' state of "society," that the fathers and mothers should more than ever protect their sons and daughters from allying themselves with those whose family are in no way suited to their own, and whose only qualification is money. After all, what is money? Surely it is dearly bought if you have to marry it, and it alone; probably there is not an idea in common with the family who possess it, on either side, father or mother; they may never even have had grandfathers, or if so, probably of very humble origin, and in no way can their offspring be suitable companions for your children for life, and very often when married in a much higher sphere they expect that you have married not only themselves, but, also, their families. But to return to "society" as it now is. What is it? A new word has cropped up within the last ten years: "smart" society. Is it recruited from blood? assuredly not. Is it exemplary virtue? assuredly not. Is it exquisite wit? No, it is rich Jews, Americans, and those who must be ''en Evidence'', and that they only can be from entertainments that alone cost far more than the very highest giving of the aristocracy of our country could or would deem it expedient to afford in so poor a cause; but the ''nouveaux riches'' have to buy their way into our present London society, and except by spending large sums this end cannot be attained. Their ostentatious display would in itself prevent, and does prevent, many of the "noble of the land" from ever encouraging their impertinent overtures to induce them to visit them or to recognize them socially in any way; but there are those who "jump" at the invitations the minute they arrive, and a ready response is sent, only too willingly. But in many instances the excuse for going to these houses is, "You know we have ''all'' our daughters to marry and those people "who give these gorgeous feasts are all so colossally rich." Are they? Not always. Ask them in view of marriage to ''settle'' a sum on your son or daughter, as the case may be, and the answer generally is, "Trust to us to make money matters all right." We know in several instances the value of these assurances. While money lasts they probably make a fair allowance to the young couple, but a crash comes, and where is ''the fair allowance'', not to speak of a "settlement,<nowiki>''</nowiki> which of course has never been made. Mothers who take their daughters to the houses of the ''nouveaux riches'', of whatever nationality, have only themselves to thank if misfortune overtakes their children eventually, if it is by marriage that they have allied themselves to such people. I know at present of three ladies in London, but not in what is now termed "society," who would not for one moment admit any one of the "new" people to their houses. Without doubt they are the most exclusive in London. Happily for them, none of them have "daughters to marry." One is the wife of an exLord Lieutenant of Ireland, and the others, two sisters of high birth and of exquisite refinement, the wives of earls and the daughters of earls. But those distinguished ladies are in the minority; the greatest compliment one can pay them is: "You 'never hear of them;' they are not 'advertising ladies.'" Many of our great ladies no longer exist. Lady William Russell, Lady Holland — where are they? Alas! no longer with us. Cleveland House is, through change of hands, no more the home of the Duchess of Cleveland, and several more hostesses, from one cause and another, entertain no longer, and their places filled — how? Why, not at all. Where is the ''grande dame'' of only a few years ago? True, there are the Embassies, and very well done are all entertainments at them. The Russian and Austrian are quite of the very best description. With such hostesses nothing else could be expected, but where are the ladies of Great Britain? Certainly not in London. Our sovereign and princes never for a moment contemplate competing with the ostentatious plutocrats of to-day. Nor even do our highest aristocracy strive to emulate them; but it might effect a change if they would set an example of aristocratic simplicity, so far as is compatible with their great position. What the ''nouveaux riches'' do not seem to understand, is that there is no true distinction in being rich, and that no ''genuine'' reverence is extended to them simply because of their wealth. One of the greatest signs of their vulgarity is the wanton and purposeless display of opulence by people who have no other possession in the whole world to recommend them. They think they are imitating the "great ones of the land," and, were it worth while, "the great ones" could rebuke them by reducing their expenditure, having fewer domestics, fewer carriages, fewer gardeners and gamekeepers; but even were those things done, I believe the lesson would be lost, and the motive be entirely misunderstood. The ducal simplicity would be ascribed either to personal meanness or to a reduced income. I am afraid it would take a great many men of birth and wealth in these days to enter into a compact to make the experiment in question, before the world at large would even observe that any new moral dogma was being put to the test. London "society" at present is immense, but exclusive "society" is small, smaller than ever; because nowadays it is obliged to discriminate more than ever, lest by accident, unawares, a member of the large London "society" finds his way into the smaller and exclusive drawing-rooms; they know their friends, and "are known by them." Many of the hostesses of the present day know not even the name of the guest the servant announces, but the most distinguished men of the day are totally unknown in the houses of the ''nouveaux riches''. A certain set of people may go, of aristocratic birth, but probably they are impecunious (if not daughters to marry), and they think there is sure to be a good cook. A foreign royalty may go, but that is by mistake; H.R.H. may have been misled as to the social status of his host, and on his second visit to London will not again make the mistake he did on first visiting our shores. Let us hope that another season we may still have the exclusive hostesses with us, and that they will entertain in their usual unostentatious and high-bred manner. The last season was broken up by the dissolution of Parliament to a certain extent, but above all by the overwhelming calamity which happened to T.R.H. the Prince and the Princess of Wales, Her Majesty the Queen, the Royal Family, and to the nation at large.<ref>Alexander Henry Wylie, "Society in 1892." ''London Society'' December 1892 (Vol. LXII): 611–614.</ref>{{rp|611–614}} </blockquote> === The Sketch: A Journal of Art and Actuality === Begun by William Ingram and Clement Shorter as an addition to the Illustrated London News, the Sketch was first edited by Clement Shorter (ed. 1893–1900). It focused on "high society and the aristocracy" (Wikipedia. "The Sketch." https://en.wikipedia.org/wiki/The_Sketch). It was printed and published by Ingram Brothers, 198, Strand, London and cost sixpence. The British Library holds a complete run, but as of August 2016, it was not part of the British Newspaper Archive; many of the volumes below were digitized and are probably held at the University of Minnesota. Google Books has some issues; I need Vol. 18, and have found the following: *Wednesday 2 August 1893, No. 27, through 25 October 1893, No. 39, Vol. III: https://books.google.com/books?id=Z3w4AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false * *Wednesday 31 October 1894, No. 92, through 23 January 1895, No. 104, Vol. VIII: https://books.google.com/books?id=lnw4AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 30 January 1895, No. 105, through 24 April 1895, No. 117, Vol. IX: https://books.google.com/books?id=1304AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 1 May 1895, No. 118, through 24 July 1895, No. 130, Vol. X: https://books.google.com/books?id=A344AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false * *Wednesday 30 October 1895, No. 144, through 22 January 1896, Vol. XII: https://books.google.com/books?id=P344AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *1896, Vol. 13: https://books.google.com/books?id=7qI6mzrUr_QC&pg=PA340&dq=the+sketch+a+journal+of+art+and+actuality+volume+18&hl=en&sa=X&ved=0ahUKEwifrq-cheDOAhUF1x4KHS1pA4IQ6AEIKDAC *Wednesday 29 April 1896, No. 170, through 22 July 1896, No. 182, Vol. XIV, plus Supplement: https://books.google.com/books?id=fH44AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 29 July 1896, No. 183, through 21 October 1897, No. 195, Vol. XV: https://books.google.com/books?id=sH44AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 28 October 1896, No. 196, Vol. XVI, through 9 December 1896, No. 202, Vol. XVI: https://books.google.com/books?id=uX44AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false * *Wednesday 28 April 1897, No. 222, through 21 July 1897, No. 234, Vol. XVIII: https://books.google.com/books?id=fQxIAQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 28 July 1897, No. 235, through 20 October 1897, No. 247, Vol. XIX: https://books.google.com/books?id=JH84AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday January 26 1898, No. 261, through 20 April 1898, No. 273, Vol. XXI: https://books.google.com/books?id=Z384AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false * *Wednesday 27 July 1898, No. 287, through 19 October 1898, No. 299, Vol. XXIII: https://books.google.com/books?id=kn84AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 26 October 1898, No. 300, through 18 January 1899, No. 312, Vol. XXIV: https://books.google.com/books?id=sn84AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false * *Wednesday 25 October 1899, No. 352, through 17 January 1900, No. 364, Vol. XXVIII: https://books.google.com/books?id=4n84AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Wednesday 24 January 1900, No. 365, through 18 April 1900, No. 377, Vol. XXIX: https://books.google.com/books?id=G4A4AQAAMAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false *Vol. XXX *Vol. XXXI *Vol. XXXII *Vol. XXXIII *Vol. XXXIV *Vol. XXXV *Vol. XXXVI *Vol. XXXVII *Vol. XXXVIII *Vol. XXXIX *Vol. XL *Vol. XLI *Vol. XLII *Vol. XLIII *Vol. XLIV *Vol. XLV *Vol. XLVI * *Vol. XLIX * *Vol. LI * *Vol. LIII *Vol. LVI === Quarterlies === * ''The Fortnightly Review'' (1865–; V. 62-63, 1894-95; V. 64-66, 1895-96) (British Library DSC Shelfmark 4018.340000) === Minor Magazines === * ''The Chameleon,'' an undergraduate literary magazine published by Oxford undergraduates. Lord Alfred Douglas's "Two Loves" was originally published in the December 1894 issue. == Resources for Working with Victorian Periodicals == === Researching the Periodicals, Authors, Etc. === * ''The Curran Index to Nineteenth-Century Periodicals'': https://www.curranindex.org/. Citing: Database: ''The Curran Index'', eds. Lars Atkin and Emily Bell. 2017-present. curranindex.org. Entry: ‘[Page Title].’ ''The Curran Index'', eds. Lars Atkin and Emily Bell. [URL], [date of access]. The Currran Index builds on the work in the ''Wellesley Index'', below. * ''The Wellesley Index To Victorian Periodicals 1824–1900''. 5 Vols. Ed., Walter E. Houghton. U of Toronto Press, Routledge & Kegan Paul, 1966. *# Volume I ([[iarchive:wellesleyindexto0001unse/|https://archive.org/details/wellesleyindexto0001unse]]) *#* ''Blackwood's Edinburgh Magazine'' *#* ''The Contemporary Review'' *#* ''The Cornhill Magazine'' *#* ''The Edinburgh Review'' (including the years 1802–1823) *#* ''The Home and Foreign Review'' *#* ''Macmillan's Magazine'' *#* ''The North British Review'' *#* ''The Quarterly Review'' *# Volume II (https://archive.org/details/wellesleyindexto0002unse) *#* ''Bentley's Quarterly Review'' *#* ''The Dublin Review'' *#* ''The Foreign Quarterly Review'' *#* ''The Fortnightly Review'' *#* ''Fraser's Magazine'' *#* ''The London Review'' (1829) *#* ''The National Review'' (1883–) *#* ''The New Quarterly Magazine'' *#* ''The Nineteenth Century'' *#* ''The Oxford and Cambridge Magazine'' (1856) *#* ''The Rambler'' (1848–1862) *#* ''The Scottish Review'' (1882–) *#Volume III (https://archive.org/details/wellesleyindexto0003unse) *#*''Ainsworth Magazine'' *#*''The Atlantis'' *#*''The British and Foreign Review'' *#*''The London Review'' (1835–1836) *#*''The London and Westminster Review'' (1836–1840) *#*''The Modern Review'' *#*''The Monthly Chronicle'' *#*''The National Review'' (1855–1864) *#*''The New Monthly Magazine'' (1821–1854) *#*''The New Review'' *#*''The Prospective Review'' *#*''Saint Pauls'' *#*''Temple Bar'' *#*''The Theological Review'' *#*''The Westminster Review'' (1824–1836, 1840–1900) *#Volume IV (https://archive.org/details/wellesleyindexto0004unse) *#*''Bentley's Miscellany'' *#*''The British Quarterly Review'' *#*''The Dark Blue'' *#*''The Dublin University Magazine'' *#*''The London Quarterly Review'' *#*''Longman's Magazine'' *#*''Tait's Edinburgh Magazine'' (1832–1855) *#*''The University Magazine'' *#Volume V, Ed., Jean Harris Slingerland (https://archive.org/details/wellesleyindexto0005unse) *#*Epitome and Index * Directories ** [Mitchell's] Newspaper Press Directory, Vol. 52. London: Messrs C. Mitchell & Co., 1897: [https://babel.hathitrust.org/cgi/pt?id=mdp.39015085486150&view=1up&seq=250&q1=%22The+Lady%22 https://babel.hathitrust.org/cgi/pt?id=mdp.39015085486150] * ''Victorian Fiction Research Guides'': https://victorianfictionresearchguides.org/ === Sources of Digitized Periodicals === * The ''British Newspaper Archive'': https://www.britishnewspaperarchive.co.uk/. The page numbering in the BNA does not match the page numbers on the printed page, and the title may not be accurate for that date, either. (e.g., 63 [of 97 in BNA; p. on print page], Col. 2a–3a [3 of 3 cols.]) * ''Google Books'' has some periodicals digitized and still available through them. * The ''Hathi Trust Digital Library'': https://www.hathitrust.org/ (accessed December 2022). * ''Internet Archive'': [[iarchive:howtodoitordire00unkngoog/page/n68/mode/2up|https://archive.org/details/]] * Library of Congress ''Chronicling America'' (for American newspapers): https://chroniclingamerica.loc.gov/ * The London ''Times'' * The ''Newspaper Archive'': https://newspaperarchive.com/ * ''The Online Books Page'' University of Pennsylvania Libraries: https://onlinebooks.library.upenn.edu/ (accessed December 2022) * ''Open Access Nineteenth-century Periodicals'', at The Victorian Web: https://victorianweb.org/periodicals/openaccess.html ==Bibliography== *[1884-03-07 Pictorial World] The Pictorial World 7 March 1874 (1:1). Old Pictorial: Press from Our Past. Online http://www.oldpictorial.com/publishedby/pictorial-world/. *[1894-04-04 Sketch 5, Col. 1C] "L. E." "A Chat with Lady Violet Greville." The Sketch 4 April 1894, Wednesday: 5, Col. 1A. (Behind paywall: http://www.britishnewspaperarchive.co.uk/viewer/bl/0001860/18940404/007/0005) Accessed December 2016. *Beetham, Margaret, and Kay Boardman, eds. Victorian Women's Magazines: An Anthology. Manchester and New York: Manchester University Press, 2001. Google Books. *Gliserman, Susan. "Mitchell's 'Newspaper Press Directory': 1846–1907." Victorian Periodicals Newsletter, No. 4 April 1969 (2: 1): 10–29. *Hindle, Wilfred. The Morning Post: 1772–1937, Portrait of a Newspaper. London: Rutledge, 1937. *Leary, Patrick. "Re: [VICTORIA] Victorian news parody." Reply to a posting on the Victoria listserv (victoria@list.indiana.edu). Monday, January 21, 2019 at 9:25 AM. *Miliband, Marion, ed. ''The [London] Observer of the Nineteenth Century, 1791-1901.'' London: Longmans, 1966. DA530.O2. *[Mitchell]. Newspaper Press Directory, Vol. 52. London: Messrs C. Mitchell & Co., 1897. *[NPD 1905] Newspaper Press Directory: And Advertisers' Guide, Containing Full Particulars of Every Newspaper, Magazine, Review, and Periodical Published in the United Kingdom and the British Isles. The Newspaper Map of the United Kingdom, the Continental, American, Indian and Colonial Papers, and a Directory of the Class Papers and Periodicals. Diamond Jubilee Issue. 60th annual issue. London: C. Mitchell and Co., 1905. ''Google Books''. https://books.google.com/books?id=mGMLAAAAYAAJ. *Onslow, Barbara. "The Ladies' Page." Victorian Page: The Web Magazine of Victoriana. Web. Accessed April 2017. http://www.victorianpage.com/VictorianPage-Ladiespage-womensmagazines.html *Sell, Henry. Sell's Dictionary of the World's Press. London, Sell's: 1886. ''Google Books''. https://books.google.com/books?id=SEsCAAAAYAAJ. *Thomas, Frederick Moy, ed. Fifty Years of Fleet Street: The Life and Recollections of Sir John Robinson. London: Macmillan, 1904. Google Books: https://books.google.com/books?id=-mMLAAAAYAAJ. *Thorold, Algar Labouchere. The Life of Henry Labouchere. New York: G. P. Putnam's, 1913. *[Who's Who 55] Addison, Henry Robert, and Charles Henry Oakes, William John Lawson, Douglas Brooke Wheelton Sladen, eds. Who's Who, 1903. 55th edition. London, Adam and Charles Black, 1903. Google Books. *Willing's British and Irish Press Guide and Advertiser's Directory and Handbook. ["Late May's."] 18th ed. n.p., 1891. Google Books. https://books.google.com/books?id=104CAAAAYAAJ. == References == {{reflist}} [[Category:Newspapers]] tr5p7uf6l9zl07f79momfoo1odls1a1 0 285384 2690940 2690743 2024-12-09T02:15:25Z Young1lim 21186 /* Linking Libraries */ 2690940 wikitext text/x-wiki === 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.20241207.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === 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.20241209.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]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] o43byia0ubtuzhg4smew1t7e4bioikc 2690991 2690940 2024-12-09T09:03:18Z Young1lim 21186 /* Integer Arithmetic */ 2690991 wikitext text/x-wiki === 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.20241209.pdf |pdf]]) ==== Floating point Arithmetic ==== </br> === 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.20241209.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]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming language]] 3lmzismrbt2y0xl6rdw0fvzs9xv1bo3 0 289706 2690970 2449344 2024-12-09T05:05:22Z 137.157.8.253 Upper case to a lower case in the middle of a sentence. 2690970 wikitext text/x-wiki ==Learning Contents Summary == The contents of '''Record Management Stages/LifeCycle Model''' as a learning resource is to expose [[w:Librarians|librarians]] and [[w:Library|library]] [[w:Students|students]] to the concept of [[w:Record management|record management]] and its various stages. This [[Course materials|course material]] serves as an avenue for [[w:Learners|learners]] from diverse [[w:Academic disciplines|academic disciplines]] to acquire [[w:Knowledge|knowledge]] on how libraries and [[w:Archival centers|archival centers]], create, process, store, disseminate, and dispose of [[w:Records|records]]. ==Goals== At the end of this [[w:Study|study]], learners should be able to: *understand the definition of records, their examples, and types; *differentiate between record management and its models; and *the activities carried out in a record lifecycle model. {{TOC right|limit|limit=2}} ==CONCEPT OF RECORD== [[File:Records Life Cycle.svg|thumb|Life Cycle of Record|220px]] A record, according to the [[w:International Foundation for Information Technology|International Foundation for Information Technology]] (2010), is something that represents proof of existence and can be used to recreate or prove the state of existence, regardless of medium or characteristics<ref>{{Cite web|url=https://if4it.com/|title=International Foundation for Information Technology – Just another WordPress site|language=en-US|access-date=2022-11-08}}</ref>. [[w:International Standard Organization|International Standard Organization]] (ISO) Standard 15849 defines a record as "recorded [[information]] created, received, and maintained as evidence by an [[organization]] or [[w:Person|person]] in pursuance of [[w:Legal obligations|legal obligations]] or the [[w:Transaction of business|transaction of business]]." It is necessary for records to accurately reflect what was communicated, decided, or what action was taken; to support the needs of the organization, and to support [[w:Accountability|accountability]]. A record consists of the outcomes of significant daily activities that support an organization's mission and objectives, such as<ref>International Organisation for Standardisation (iso) (2001). ISO 15489-1 Information and documentation-records management: Part 1 general. Geneva: ISO.</ref>; *advice and recommendations made to management and the decisions and actions taken as a result, along with supporting [[w:Documentation|documentation]]; *problems encountered in [[w:Organizational operations|organizational operations]] and the steps are taken to resolve the problems; *interactions with the [[w:Public|public]], [[w:Customers|customers]], [[w:Clients|clients]], [[w:Stakeholders|stakeholders]], [[w:Consultants|consultants]], [[w:Vendors|vendors]], [[w:Partners|partners]], and other [[w:Government|government]] jurisdictions; *[[w:Verbal communications|verbal communications]] such as [[w:Meetings|meetings]], [[w:Telephone calls|telephone calls]], and [[w:Face-to-face discussions|face-to-face discussions]] where significant actions or decisions have occurred; *[[w:Legal agreements|legal agreements]] of any kind, including [[w:Contracts|contracts]], along with supporting documentation; *[[w:Policy|policy]], [[w:Organizational planning|organizational planning]], [[w:Performance measurement|performance measurement]], budget activities, and supporting documentation; *work done for the government by [[w:Consultants|consultants]] and other external resources; and *actions and decisions where payments are made or received, [[w:Funds|funds]] committed, services delivered, or [[w:Obligations|obligations]] incurred. A record can be [[w:Correspondence|correspondence]], a [[w:Memorandum|memorandum]], a [[w:Book|book]], a [[w:Plan|plan]], a [[w:Map|map]], a [[drawing]], a [[Diagram drawing|diagram]], a [[w:Pictorial or graphic work|pictorial or graphic work]], a [[Photography as art|photograph]], a [[w:Film|film]], [[w:Microfilm|microfilm]], a [[w:Sound recording|sound recording]], a [[w:Videotape|videotape]], a [[w:Machine-readable record|machine-readable record]], or any other [[w:Documentary material|documentary material]], regardless of physical form or characteristics<ref>Pearce-Moses, R. (2005). A glossary of archival and records terminology. Chicago, Il: The society of American archivists. p. 23 </ref>. ==RECORD LIFE CYCLE== [[w:Records life-cycle|Records life-cycle]] in records management refers to the stages of a record's ''life span'' from its [[w:Creation|creation]] to use, [[w:Maintenance|maintenance]], [[w:Preservation|preservation]], and or [[w:Disposal|disposal]]. [[w:IBM Corporation|IBM Corporation]] (2009) describes the record life cycle as to how long a record is kept; what actions should be taken (if any) as the record progresses through various stages of [[w:Retention|retention]]; and what happens to the record when its life cycle is complete. It is a series of stages through which a record must pass before being disposed off. A life cycle, for example, can be as short as [[w:Zero|zero]] days or have no defined end. Each life cycle phase has a specific duration and denotes a specific records management activity that a records manager performs at the beginning or end of the phase. The duration of life cycles varies, and records management policy determines the duration of records.<ref>{{Cite web|url=https://www.ibm.com/docs/en/rmfz/8.5.0?topic=schedules-record-life-cycles-overview|title=Record life cycles overview|website=www.ibm.com|language=en-us|access-date=2022-11-08}}</ref> ==RECORD MANAGEMENT STAGES/ LIFECYCLE MODEL== According to Roper (1977) the earliest verified stages of the record’s life cycle are divided into three stages that include active or current; intermediate or semi-current, and finally archives or non-current<ref>Roper, M. (1977). This is records management. in: records management conference, 21 oct. [s.l.]: [s.n.].</ref>. These three stages in the life of a record are basic to any records management program. The three stages of the life cycle are also identified by Hardcastle (1989: 60)<ref>Hardcastle, S. (1989). Providing storage facilities. in: Peter Emmerson (ed.). How to manage your records: A guide to effective practice. Cambridge ICSA publishing</ref>. These stages include the current stage when the records are active; the non-current stage when the records are inactive; and the archival stage when records are useful for historical rather than business purposes<ref>Mokhtar, Umiasma’ (2017). Records classification: Concepts, principles and methods records management models. 63–96. doi:10.1016/b978-0-08-102238-2.00004-4 </ref>. The three-stage vision of the record’s life cycle was also shared by Charman (1984: 2) who stated that the life cycle of a record is its progression from creation to the final disposal.<ref>Charman, d. (1984). records surveys and schedules: a ramp study with guidelines. Paris: UNESCO.</ref><ref>Tayfun, A.C. & Gibson, S.A. (2022). Model for life cycle records management, article, October 1, 1996; Vienna, Virginia. <nowiki>https://digital.library.unt.edu/ark:/67531/metadc681427/</nowiki>: university of north Texas libraries, UNT digital library, <nowiki>https://digital.library.unt.edu</nowiki>; crediting UNT libraries government documents department.</ref> It includes the following phases: ===Active or Current Records=== Records that are regularly used for the current business of an agency or organization and continue to be maintained in their place of origin or receipt; records in this stage are sometimes called active records. For example, if you went to the dentist last week or even a few months ago, then your record would be considered active. However, if you last visited your [[w:Dentist|dentist]] over seven years ago, then your record may be considered inactive. ===Intermediate or Semi-Current Records=== Kevin Ashley from the [[w:Digital Curation Centre|Digital Curation Centre]] (2022) described them as ‘the undead’- records that are still with us but not quite alive. Records that are required so infrequently for current business or activities that they should be transferred to a records center pending their ultimate disposal. Example of such records includes; closing equipment inventory, [[w:Financial summary|financial summary]], demonstrations, exhibits, [[w:Leadership|leadership]] and [[w:Citizenship|citizenship]], project highlights, [[w:Auctioneer|auctioneer]] statement, and [[w:Club|club]] meeting log among others. ===Non-Current or Inactive Records=== Records that are no longer required for current business should be either destroyed or transferred to an archival repository. For example, many [[Colleges and Universities|colleges]] are required to keep records of students, although the student may have attended decades ago. Many of these inactive records are required to be kept for legal, administrative, or even historical reasons. Non-records or transitory records are not required to control, support, or document the delivery of programs, carry out operations, make decisions, or account for activities of the department. Non-records should be managed and routinely disposed of properly once the administrative, legal or fiscal use has expired, (RIM taskforce, 2008)<ref>RIMtaskforce (2008). Guidelines for determining what records need to be retained. Retrieved from:<nowiki>https://www.pimedu.org%2ffiles%2ftoolkit%2fpimrisk1.pdf&clen=201409&chunk</nowiki></ref>. These may include: *'''Advertising materials''': solicited or unsolicited information you receive from businesses or individuals advertising their products or services. *'''Blank information media''': blank information media, e.g., [[w:Letterhead|letterhead]], blank [[w:CDs|CDs]], etc. *'''Draft documents and working materials''': correspondence, reports, and other documents, which usually have not yet been finalized. These include research or working materials such as calculations and notes that are often collected and used in the preparation of documents. Once the final version of a document is complete and filed, most drafts and working materials should be disposed of. *'''Duplicate copies;''' where nothing has been added changed or deleted, where the copy is used for information or reference only, and where the original is filed in the records management system. *'''External publications''': books, [[w:Magazines|magazines,]] [[w:Periodicals|periodicals]], [[w:Pamphlets|pamphlets]], [[w:Brochures|brochures]], [[journals]], [[w:Newspapers|newspapers]], and software documentation, whether printed or electronic, that you have obtained from sources outside your organization. Publications that are about schools or school boards/authorities may have historical value and should be retained as part of the records management program. *'''Routine notices''': notices that contain information useful for only a brief period, after which it has no further value or is of little interest. Note that the originating department is responsible for retaining the notice if it supports departmental activities, responsibilities, or communication. *'''Information of short-term value/unsolicited information''': information received as part of a distribution list, or [[w:Email messages|email messages]] received from [[w:Listservs|listservs]] and other [[Internet]] sources, solely for convenience of [[w:Reference|reference]]. The number of stages in the life cycle of records advocated by Wallace, Lee, and Schubert (1992) extends beyond those discussed above<ref>Wallace, P. E., Lee,  J.  & Schubert, D. T. (1992). Records management integrated information systems. 3rd ed. Englewood liffs, New Jersey, prentice hall.</ref>. According to them, the lifecycle of a record is divided into seven stages, which include: ===The Creation=== The creation of records according to the Ohio State University (2022) is also known as a receipt. It is the first stage of a record's life<ref>{{Cite web|url=https://library.osu.edu/osu-records-management/lifecycle|title=Records Lifecycle {{!}} Ohio State University Libraries|website=library.osu.edu|access-date=2022-11-08}}</ref>. It refers to the process by which a record is created within an organization, including but not limited to: *typing/word processing of a document *typing and sending an email *construction of a [[Spreadsheets|spreadsheet]] *recording of a meeting *entering a transaction within an enterprise system *the receipt of documents *the receipt of spreadsheets *the receipt of the email Records are created or received, throughout the creation phase by the process of collecting and capturing information in various forms such as paper-based records, microform-based records, and electronic-based records, (coursehero.com, 2022)<ref>{{Cite web|url=https://www.coursehero.com/file/41686787/THE-RECORDS-LIFE-CYCLE-CONCEPTpdf/|title=The records life cycle concept|date=2022|website=www.coursehero.com|access-date=2022-11-08}}</ref>. At this stage, the records are very active. ===Records distribution=== This is the second stage of the record lifecycle. After a record is created or received, it passes through a distribution phase. The distribution in the life cycle of a record includes both internal and external distribution and the impact on the entire or a portion of a business. The record is widely used during this phase and must be maintained in an easily accessible location for easy access and use. The record might be kept for a few hours or years, depending on the retention schedule, (Information Management Simplified, 2021).<ref>{{Cite web|url=https://theecmconsultant.com/what-is-records-lifecycle/|title=What is Records Lifecycle: The Complete Guide|last=Malak|first=Haissam Abdul|date=2021-12-30|website=Information Management Simplified|language=en-us|access-date=2022-11-08}}</ref> ===Records utilization=== The created desired records may be retrieved and delivered to the specified person on request for the efficient disposal of business functions. It involves the development of specified procedures through which records move from one person to another. As information is only valuable if it can reach multiple audiences at the proper moment. ===Records storage=== This stage involves the proper filing and care for the records. While many records may be disposed of after their initial use, others are required to be kept for a longer period for legal, fiscal, or other administrative reasons. Since immediate access to these records is no longer required during this phase, they are typically stored online, offsite or offline so as not to burden the operating office's storage capacity or the operating system's efficiency. The records are properly classified and put into separate file covers or folders. A proper filing system should be followed for keeping documents and should be stored in an easily accessible place. Proper care should be taken to protect every record, as maintenance of records is very much essential for effective management. This stage is further divided into two, namely; ====Active storage==== when you store a paper record in a file cabinet close to your workstation or an electronic document in an easily accessible file location or when an electronic record is considered active within a large database system, (New York State Archive, 2021)<ref name=":0" />. ====Inactive storage==== When you stop regularly referring to a record, and no longer need immediate access to it, the record enters the inactive phase. Although you may never refer to the record again, the record must be retained for legal, administrative, or other purposes. At this point, you should remove the records from their active storage environment and maintain the records in a secure records storage facility or inactive records filing system for electronic records or indicate their inactive status within a [[Databases|database system]], (New York State Archive, 2021)<ref name=":0">New York State Archive, (2021). The record lifecycle. Retrieved from <nowiki>http://www.archives.nysed.gov/common/archives/files/the-records-lifecycle-2021-05_dlowry_08.02.2021.pdf</nowiki></ref>. ===Records transfer=== This is the fifth stage in the record management cycle. It is the stage whereby records are either transferred to the archive for retention or disposed of. According to Public Record Office Victoria (PROV) (2021), it entails taking custody, but not ownership, of permanent public records from an agency. Record is transferred when appraised as permanent using the appropriate Retention and Disposal schedule to reduce storage space and cost of maintenance<ref>Public Record Office Victoria (prov) (2021). What is a record transfer? Retrieved from <nowiki>https://prov.vic.gov.au/recordkeepinggovernment/transferringrecords#:~:text=record%20transfer%20is%20a%20process,from%20a%20victorian%20government%20agency</nowiki></ref>. If records are to be transferred to the archival institution, the following steps are to be taken; *Use the action dates file to identify records that are scheduled to be transferred to the archival institution. *Remove these boxes and check the contents to ensure that all metal pins, clips, etc have been removed and the records are clean and in order. *Amend the relevant transfer list in the master transfers file for the agency concerned. *Remove the copy of the transfer list from the action dates file and place this copy in a record transferred to the archives file. *All records to be transferred to the archival institution must be listed. The archival institution will likely use an accession form, and it may ask that this form be completed when preparing the records for transfer. *Notify the originating agency of the change in the status of the records by sending them an amended copy of the transfer list. *As records transferred to the archival institution become subject to different regulations, refer any future requests for access to these records to that institution. ===Records disposition=== According to US National Archive (2022) Disposition refers to those actions taken regarding records after they are no longer needed in office space to conduct current agency business or activities.<ref>{{Cite web|url=https://www.archives.gov/records-mgmt/scheduling/rdo|title=Records Disposition Overview|date=2018-02-05|website=National Archives|language=en|access-date=2022-11-08}}</ref> These actions include: *Transfer of records to agency storage facilities or records centers. *Transfer of records from one agency or department to another. *Transfer of permanent records to the Archives of an institution for preservation and research. *Disposal of temporary records no longer needed to conduct business, usually by destruction or occasionally by donation. The records and documents no longer required are destroyed after getting approval from top management. Obsolete and unnecessary records are destroyed to avoid needless storage costs and avoid storage space also. These are also informed to the top management. Valuable documents such as deeds, bonds, registration certificates, tax returns, property documents, and the like are retained for future use and permanent storage. In contrast to disposition, ‘’disposal’’ in Federal usage refers to only those final actions taken regarding temporary records after their retention periods expire. It normally means the destruction of the record content. The term is also used occasionally to mean the transfer of temporary records from Federal control by donating them to an eligible person or organization after receiving the record manager’s approval. Destruction is accomplished in a variety of ways including, but not limited to: *disposal in trash or [[w:Recycling bin|recycling bin]] *shredding *[[w:Incineration|incineration]] *deleting an [[w:Electronic file|electronic file]] *shredding of [[w:Optical disk|optical disk]] *burning the recording medium ==Questions for Practice== Print out this section and provide the correct answers to the questions. #What is a Record? #What are the types of Records known to you? #List the stages of the record lifecycle and briefly explain 3. #What is the difference between disposition and disposal of record? ==REFERENCES== <references /> [[Category:Library and Information Science stubs]] [[Category:Library and Information Science]] [[Category:Information Organization/Encoded Archival Description]] a5ha6p3seys7nyqmyhdkizahq20qzzn 118 298255 2690971 2690653 2024-12-09T05:05:50Z Jaredscribe 2906761 /* Further Reading and Commentary suggested by Wikiversity Professors */ author links 2690971 wikitext text/x-wiki <!--{{Infobox book | | name = Aristotle for Everybody: Difficult Thought Made Easy | image = File:Aristotle for Everybody, first edition.jpg | caption = Cover of the first edition | author = [[Mortimer J. Adler]] | country = United States | language = English | subject = [[Aristotle]] | publisher = [[Macmillan Publishers]] | pub_date = 1978 | media_type = Print ([[Hardcover]] and [[Paperback]]) | pages = 206 (paperback edition) | isbn = 978-0684838236 }}--> An Introduction to Common Sense and [[w:Aristotelianism|Aristotelian philosophy]] on wikisource, arranged according to Aristotle's list of intellectual virtues in Book VI of the [[w:Nicomachean_Ethics|Nicomachean Ethics]]. * Productive reason * Practical reason * Theoretical reason Selected chapters compiled by {{cite book|title=Aristotle for Everybody: Difficult Thought Made Easy|title-link=Aristotle for Everybody|last=Adler|first=Mortimer|date=1997|publisher=Touchstone|ISBN=0-684-83823-0|location=New York|author-link=w:Mortimer J. Adler|orig-date=1978}} This is is his "Epilogue: for those who have read or wish to read to [[w:Aristotle|Aristotle]]", described as a "second table of contents" to his book, with a set of titles that "more precisely describes the Aristotelian doctrines being expounded in the five parts of [Adler's] book". Under the title of each of its twenty three chapters, there is are ''"brief statements, in Aristotelian language, of the doctrines being expounded in that chapter"'', ''which will printed in italics,'' with original research in plain font. The "list of references to appropriate portion's of Aristotle's works" were compiled by professor Adler.{{sfn|Adler|1978|loc=Epilogue, pp.192-193}} We accept his invitation to "check [his] exposition" using this reading companion "against the texts on which [he] has relied for the main tenets of Aristotle's thought"{{sfn|Adler|1978|loc=Epilogue, pp.192-193}}, and encourage other students to acquire and read Adler's book, and contribute to this project. For use as an undergraduate course on "Introduction to Philosophy" or "Introduction to Aristotle". == Part I: Man the Philosophical Animal == === 1. Aristotle's Fourfold Classification of Sensible, Material Substances: Inorganic Bodies, Plants, Animals, Men. [Philosophical Games] === ''The criteria by which Aristotle distinguished between living and non-living things; within the domain of living things, between plants and animals, and within the domain animal life, between brute animals and [[w:Rational_animal|rational animal]]<nowiki/>s, id est, human beings'' / [[w:Homo_sapiens|homo sapiens]]. Division by [[w:Dichotomies|dichotomies]] that are [[w:Jointly_exhaustive|jointly exhaustive]] and [[w:Mutually_exclusive|mutually exclusive]]. * [[w:Metaphysics (Aristotle)|Metaphysics]] [[s:Metaphysics (Ross, 1908)/Book 1|I.1]] * [[w:On the Soul|On the Soul]] [[s:On the Vital Principle/Book 1/Chapter 1|I.1]] [[s:On the Vital Principle/Book 1/Chapter 5|I.5]]; [[s:On the Vital Principle/Book 2/Chapter 1|II.1-3]],5,9; [[s:On the Vital Principle/Book 3/Chapter 1|III.3]],12 * [[w:History of Animals|History of Animals]] [[s:History_of_Animals_(Thompson)/Book_X|X.1]] * [[w:Generation of Animals|Generation of Animals]] I.1-9 IV.4-6 * [[w:Parts of Animals|Parts of Animals]] I.4-5 ''Aristotle was aware of difficulties in applying this scheme of classification. The difficulties arise because of the existence of borderline cases that straddle the lines that divide the living from the nonliving, and plants from animals.'' * [[w:History of Animals|History of Animals]] [[s:History_of_Animals_(Thompson)/Book_VIII|VIII.1]] ''The difference between [[w:Essence|essential]] and [[w:Accident_(philosophy)|accidental]] differences.'' * [[w:Categories (Aristotle)|Categories]] V * [[w:Metaphysics (Aristotle)|Metaphysics]] V.4,11; IX.8 === 2. The Range of Beings: The Ten Categories [The Great Divide] === ''The being of objects that do not exist in the way that sensible, material substances exist (e.g. mathematical objects, fictions, minds, ideas, [[immaterial substance]]<nowiki/>s, such as the disembodies intelligences that are the celestial motors, and God).'' [[w:Ousia|Ousia]] * Metaphysics III.5-6, XII.8, XIII.1-5 * On the Heavens, II.1, II.12 * On the Soul, III.4-6 ''The distinction between substance and accident, id est, between bodies and their attributes.'' * Categories 5-7 * Physics I.2 * Metaphysics VII.4-6 ''The foregoing distinction is related to the point that material substances are the subjects of change, and their accidents are the respects in which they change.'' * Physics I.6-7, II.3 ''Essence or specific nature in relation to substantial form.'' * Metaphysics, V.4, V.11, VII.16-, VIII1-6, IX.8 * On the Soul II.4 ''The hierarchy of specific natures or essences'' * Metaphysics VIII.3 * On the Soul, II.3 ''Aristotle's inventory of the various categories under which the accidental attributes of substance fall.'' * Categories 4 ''Among the accidents of substance, some are permanent or unchanging; these are the properties that are inseparable from the essential nature of each kind of material substance.'' * Topics V.1-3 ''Aristotle's policy with regard to the ambiguity of words.'' * On interpretation, 1 * Topics II.4 === 3. Productive, Practical, and Theoretic Reason or Mind [Man's Three Dimensions] === ''Aristotle's threefold division of intellectual activity or thought, into thought for the sake of making things, thought for the sake of moral and political action, and thought for the sake of acquiring knowledge as an end in itself.'' Aristotle's classification of three activities of a human being: making, doing, and knowing, corresponding to the three types of reason: productive, practical, and theoretical. Adler titles these sections "Man the Maker," "Man the Doer," and "Man the Knower," respectively. * Ethics VI.2-4 * On the Soul, III.7 == Part II: Aristotle's philosophy of Nature and of Art. [Man the Maker] == {{expand section|date=April 2021}} In response to the errors and partial truths of: * [[w:Parmenides|Parmenides]] and his disciple [[w:Zeno of Elea|Zeno of Elea]] * [[w:Heraclitus|Heraclitus]] and his disciple [[w:Cratylus|Cratylus]] Aristotle developed his theory of change. It involves distinction between [[w:inertia|inertia]] (or rest) and movement. In local motion, there is a distinction between natural movement and violent or [[w:projectile motion|projectile motion]]. There is also change in quality, such as when a green tomato ripens and becomes red. This type of change can be either natural or artificial, for example a green chair can be painted red. There can be a change in quantity. There can also be [[w:generation and corruption|generation and corruption]] - coming to be and passing away. Aristotle takes note of what we now call [[w:conservation of matter|conservation of matter]]. === 4. Nature as artist and the human artist as imitator of nature === ''The difference between what happens by nature and what happens by art.'' * Physics I.7-8; II.1-3, II.8-9 * Poetics 1-4 ''The difference between what happens by art and what happens by chance.'' * Physics II.4-6 * Politics I.11 ''The difference between the changes brought about by nature and the changes brought about by art'' * Metaphysics VII.7-9 ''The difference between man's production of corporeal things and the generation or procreation of living things in nature.'' * Generation of Animals * Metaphysics VII.7 === 5. Three main modes of accidental change: change of place, change of quality, change of quantity === === 6. Aristotle's doctrine of the [[w:four causes|four causes]]: efficient, material, formal, and final. === Physics, [https://el.wikisource.org/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CE%AE%CF%82_%CE%91%CE%BA%CF%81%CE%BF%CE%AC%CF%83%CE%B5%CF%89%CF%82/2#%CE%9A%CE%B5%CF%86%CE%AC%CE%BB%CE%B1%CE%B9%CE%BF_3 II.3-9] Metaphysics [[wikisource:Metaphysics_(Ross,_1908)/Book_1|I]].3-10, [[wikisource:Metaphysics_(Ross,_1908)/Book_5|V]].3, VI.2-3, VII.17, VIII.2-4, IX.8, XII.4-5 === 7. Further developments in the theory of [[w:Potentiality and actuality|Potentiality and Actuality]], and Matter and Form, especially with respect to [[w:substantial change|substantial change]], or Generation and Corruption. [To Be or Not to Be] === === 8. Aristotle's analysis of the intellectual factors in artistic production and his classification of the arts [Productive ideas and know-how] === [[w:Aristotelian physics|Aristotelian physics]] [[w:Theory of impetus|Theory of impetus]] == Part III: Ethics and Politics [Man the Doer] == === 9. The End as the First Principle in Practical thinking and the Use of Means as the Beginning of Action: The End as First in the Order of Intention and Last in the Order of Execution === ''The good as the desirable and the desirable as the good'' * Nicomachean Ethics I.1-2 ''The distinction between ends and means as good desirable for their own sake and goods desirable for the sake of something else.'' * Ethics I.5, I.7, 1.9 ''The ultimate end in practical thinking compared with axioms or self-evident truths in theoretical thinking.'' * Posterior Analytics I.2 === 10. Happiness Conceived as That Which Leaves Nothing to Be Desired and, as so Conceived, the Final or Ultimate End to Be Sought === ''The distinction between living and living well.'' * Politics I.1-2, I.9 ''The conception of happiness as a whole good life, together with various views held by individuals concerning what a good life consists in'' * Ethics I.4-5, I.7-10; X:2, X:6-8 === 11. Distinction Between Real and Apparent Goods, or Between Goods that Ought to Be Desired and Goods That are in Fact Desired, Together with Distinction between Natural and Acquired Desires === === 12. Real Goods === === 13. Moral Virtue and Good Fortune === === 14 Obligations of the Individual === === 15. Role of the State in Abetting or Facilitating the Individual's Pursuit of Happiness === {{Expand section|date=April 2021}} == Part IV: Psychology, Logic, and Theory of Knowledge [Man the Knower] == === 16. The Senses and the Intellect: Perception, Memory, Imagination, and Conceptual Thought === ''Language in relation to thought.'' * Categories I * On Interpretation, 1-2 ''Account of the external senses and of their distinction from the interior senses: the common sense, memory, and imagination.'' * On the Soul II.5-12, III.1-3 * Sense and the Sensible * History of Animals IV.8 ''Distinction between mere sensations and perceptual experience'' * Metaphysics I.1 ''Doctrine that sensations and ideas, taken by themselves or in isolation, are neither true nor false'' * Categories 4 * On Interpretation 1 * On the Soul II.6, III.3, III.6 * Metaphysics IV.5, V.29 === 17. [[w:Immediate_Inference|Immediate Inference]] and [[w:Syllogistic_Reasoning|Syllogistic Reasoning]] === ''The [[w:Law_of_contradiction|law of contradiction]] as an ontological principle and as a rule of thought.'' * On Interpretation 6 * Prior Analytics II.17 * Posterior Analytics I.11 * Metaphysics IV.3-8; IX.5-6 ''The [[w:Square_of_opposition|square of opposition]]: contradictories, contraries, and subcontraries''. Subalterns. * On Interpretation 6, 10 * Categories 10 * Prior Analytics I.2 === 18. Theoretical and Practical Truth === === 19. Theory of Knowledge and Distinction between Knowledge and Right Opinion === {{Expand section|date=April 2021}} == Part V: Difficult Philosophical Questions == {{Expand section|date=April 2021}} == Fair use of a Derivative and Copyrighted Work derived from Primary Sources in the Public Domain == Adler's epilogue is a derivative work of Aristotle's, which are in the public domain. His "Epilogue/second table of contents" is a "selection and arrangement" (and as such may be copyrighted). Reproducing it is necessary to the project to analyzing and evaluating the book, reproducing the TOC is likely to improve rather than hurt sales of the copyrighted book. Furthermore, the original being a derivative of public domain works, it is consistent with the spirit and intent of the author to create further derivative works, and furthermore he has specifically invited his readers and students to "check his exposition" against the originals. For all these reasons and more, I content that this companion course is "fair use" of Adler's "second table of contents" to his book. * [[w:User:Jaredscribe/WV:Toc_is_fair_use|User:Jaredscribe/WV:Toc is fair use]] And if non-free, and if not allowed under fair use: [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] To host this companion course on Wikiversity would honor this work in particular, would not harm but likely improve sales of his book, and would honor his lifework in general, by its harmony with the manifestos for democratic "public education" in the liberal arts, published by professor Adler and his colleagues in "The [[w:Paideia_Proposal|Paideia Proposal]]" and elsewhere. That there be no doubt, the editors request that Simon & Schuster, Touchstone Book make a [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. This course is being migrated from the wikipedia article [[w:Aristotle_for_Everybody|Aristotle for Everybody]], as it is more appropriate to this project than that. ==Notes== {{reflist}} == Further Reading and Commentary suggested by Wikiversity Professors == {{refbegin}} * {{cite book|title=Aristotle for Everybody: Difficult Thought Made Easy|title-link=Aristotle for Everybody|last=Adler|first=Mortimer|date=1997|publisher=Touchstone|ISBN=0-684-83823-0|location=New York|author-link=w:Mortimer J. Adler|orig-date=1978}} * {{cite book|title=The Basic Works of Aristotle|author=Aristotle|date=1941|publisher=Random House|editor=Richard McKeon|editor-link=w:Richard McKeon|location=New York|author-link=Aristotle}} * {{Cite book|title=Aristotle: A Very Short Introduction|last=Barnes|first=Jonathan|publisher=Oxford University Press|year=2000|isbn=978-0-19-285408-7|author-link=w:Jonathan Barnes}} {{refend}} === Classic and Scholastic Expositors === *[[s:Isagoge (Owen)|Isagoge]] (Introduction) to Aristotle's [[s:Organon_(Owen)/Categories|Categories]] by [[w:Porphyry]] ''not recommended'' but of significant influence in late antiquity through middle ages. *[[s:The_Guide_for_the_Perplexed_(1904)|Guide to the Perplexed]], by [[w:Maimonides|Maimonides]] ''recommended'': its latin translation initiated the [[w:Scholastic|Scholastic]] movement that created the medieval universities. r3ixxmcze4b0lk2pl3rpx5k1w292w21 2690972 2690971 2024-12-09T05:07:10Z Jaredscribe 2906761 /* Fair use of a Derivative and Copyrighted Work derived from Primary Sources in the Public Domain */ 2690972 wikitext text/x-wiki <!--{{Infobox book | | name = Aristotle for Everybody: Difficult Thought Made Easy | image = File:Aristotle for Everybody, first edition.jpg | caption = Cover of the first edition | author = [[Mortimer J. Adler]] | country = United States | language = English | subject = [[Aristotle]] | publisher = [[Macmillan Publishers]] | pub_date = 1978 | media_type = Print ([[Hardcover]] and [[Paperback]]) | pages = 206 (paperback edition) | isbn = 978-0684838236 }}--> An Introduction to Common Sense and [[w:Aristotelianism|Aristotelian philosophy]] on wikisource, arranged according to Aristotle's list of intellectual virtues in Book VI of the [[w:Nicomachean_Ethics|Nicomachean Ethics]]. * Productive reason * Practical reason * Theoretical reason Selected chapters compiled by {{cite book|title=Aristotle for Everybody: Difficult Thought Made Easy|title-link=Aristotle for Everybody|last=Adler|first=Mortimer|date=1997|publisher=Touchstone|ISBN=0-684-83823-0|location=New York|author-link=w:Mortimer J. Adler|orig-date=1978}} This is is his "Epilogue: for those who have read or wish to read to [[w:Aristotle|Aristotle]]", described as a "second table of contents" to his book, with a set of titles that "more precisely describes the Aristotelian doctrines being expounded in the five parts of [Adler's] book". Under the title of each of its twenty three chapters, there is are ''"brief statements, in Aristotelian language, of the doctrines being expounded in that chapter"'', ''which will printed in italics,'' with original research in plain font. The "list of references to appropriate portion's of Aristotle's works" were compiled by professor Adler.{{sfn|Adler|1978|loc=Epilogue, pp.192-193}} We accept his invitation to "check [his] exposition" using this reading companion "against the texts on which [he] has relied for the main tenets of Aristotle's thought"{{sfn|Adler|1978|loc=Epilogue, pp.192-193}}, and encourage other students to acquire and read Adler's book, and contribute to this project. For use as an undergraduate course on "Introduction to Philosophy" or "Introduction to Aristotle". == Part I: Man the Philosophical Animal == === 1. Aristotle's Fourfold Classification of Sensible, Material Substances: Inorganic Bodies, Plants, Animals, Men. [Philosophical Games] === ''The criteria by which Aristotle distinguished between living and non-living things; within the domain of living things, between plants and animals, and within the domain animal life, between brute animals and [[w:Rational_animal|rational animal]]<nowiki/>s, id est, human beings'' / [[w:Homo_sapiens|homo sapiens]]. Division by [[w:Dichotomies|dichotomies]] that are [[w:Jointly_exhaustive|jointly exhaustive]] and [[w:Mutually_exclusive|mutually exclusive]]. * [[w:Metaphysics (Aristotle)|Metaphysics]] [[s:Metaphysics (Ross, 1908)/Book 1|I.1]] * [[w:On the Soul|On the Soul]] [[s:On the Vital Principle/Book 1/Chapter 1|I.1]] [[s:On the Vital Principle/Book 1/Chapter 5|I.5]]; [[s:On the Vital Principle/Book 2/Chapter 1|II.1-3]],5,9; [[s:On the Vital Principle/Book 3/Chapter 1|III.3]],12 * [[w:History of Animals|History of Animals]] [[s:History_of_Animals_(Thompson)/Book_X|X.1]] * [[w:Generation of Animals|Generation of Animals]] I.1-9 IV.4-6 * [[w:Parts of Animals|Parts of Animals]] I.4-5 ''Aristotle was aware of difficulties in applying this scheme of classification. The difficulties arise because of the existence of borderline cases that straddle the lines that divide the living from the nonliving, and plants from animals.'' * [[w:History of Animals|History of Animals]] [[s:History_of_Animals_(Thompson)/Book_VIII|VIII.1]] ''The difference between [[w:Essence|essential]] and [[w:Accident_(philosophy)|accidental]] differences.'' * [[w:Categories (Aristotle)|Categories]] V * [[w:Metaphysics (Aristotle)|Metaphysics]] V.4,11; IX.8 === 2. The Range of Beings: The Ten Categories [The Great Divide] === ''The being of objects that do not exist in the way that sensible, material substances exist (e.g. mathematical objects, fictions, minds, ideas, [[immaterial substance]]<nowiki/>s, such as the disembodies intelligences that are the celestial motors, and God).'' [[w:Ousia|Ousia]] * Metaphysics III.5-6, XII.8, XIII.1-5 * On the Heavens, II.1, II.12 * On the Soul, III.4-6 ''The distinction between substance and accident, id est, between bodies and their attributes.'' * Categories 5-7 * Physics I.2 * Metaphysics VII.4-6 ''The foregoing distinction is related to the point that material substances are the subjects of change, and their accidents are the respects in which they change.'' * Physics I.6-7, II.3 ''Essence or specific nature in relation to substantial form.'' * Metaphysics, V.4, V.11, VII.16-, VIII1-6, IX.8 * On the Soul II.4 ''The hierarchy of specific natures or essences'' * Metaphysics VIII.3 * On the Soul, II.3 ''Aristotle's inventory of the various categories under which the accidental attributes of substance fall.'' * Categories 4 ''Among the accidents of substance, some are permanent or unchanging; these are the properties that are inseparable from the essential nature of each kind of material substance.'' * Topics V.1-3 ''Aristotle's policy with regard to the ambiguity of words.'' * On interpretation, 1 * Topics II.4 === 3. Productive, Practical, and Theoretic Reason or Mind [Man's Three Dimensions] === ''Aristotle's threefold division of intellectual activity or thought, into thought for the sake of making things, thought for the sake of moral and political action, and thought for the sake of acquiring knowledge as an end in itself.'' Aristotle's classification of three activities of a human being: making, doing, and knowing, corresponding to the three types of reason: productive, practical, and theoretical. Adler titles these sections "Man the Maker," "Man the Doer," and "Man the Knower," respectively. * Ethics VI.2-4 * On the Soul, III.7 == Part II: Aristotle's philosophy of Nature and of Art. [Man the Maker] == {{expand section|date=April 2021}} In response to the errors and partial truths of: * [[w:Parmenides|Parmenides]] and his disciple [[w:Zeno of Elea|Zeno of Elea]] * [[w:Heraclitus|Heraclitus]] and his disciple [[w:Cratylus|Cratylus]] Aristotle developed his theory of change. It involves distinction between [[w:inertia|inertia]] (or rest) and movement. In local motion, there is a distinction between natural movement and violent or [[w:projectile motion|projectile motion]]. There is also change in quality, such as when a green tomato ripens and becomes red. This type of change can be either natural or artificial, for example a green chair can be painted red. There can be a change in quantity. There can also be [[w:generation and corruption|generation and corruption]] - coming to be and passing away. Aristotle takes note of what we now call [[w:conservation of matter|conservation of matter]]. === 4. Nature as artist and the human artist as imitator of nature === ''The difference between what happens by nature and what happens by art.'' * Physics I.7-8; II.1-3, II.8-9 * Poetics 1-4 ''The difference between what happens by art and what happens by chance.'' * Physics II.4-6 * Politics I.11 ''The difference between the changes brought about by nature and the changes brought about by art'' * Metaphysics VII.7-9 ''The difference between man's production of corporeal things and the generation or procreation of living things in nature.'' * Generation of Animals * Metaphysics VII.7 === 5. Three main modes of accidental change: change of place, change of quality, change of quantity === === 6. Aristotle's doctrine of the [[w:four causes|four causes]]: efficient, material, formal, and final. === Physics, [https://el.wikisource.org/wiki/%CE%A6%CF%85%CF%83%CE%B9%CE%BA%CE%AE%CF%82_%CE%91%CE%BA%CF%81%CE%BF%CE%AC%CF%83%CE%B5%CF%89%CF%82/2#%CE%9A%CE%B5%CF%86%CE%AC%CE%BB%CE%B1%CE%B9%CE%BF_3 II.3-9] Metaphysics [[wikisource:Metaphysics_(Ross,_1908)/Book_1|I]].3-10, [[wikisource:Metaphysics_(Ross,_1908)/Book_5|V]].3, VI.2-3, VII.17, VIII.2-4, IX.8, XII.4-5 === 7. Further developments in the theory of [[w:Potentiality and actuality|Potentiality and Actuality]], and Matter and Form, especially with respect to [[w:substantial change|substantial change]], or Generation and Corruption. [To Be or Not to Be] === === 8. Aristotle's analysis of the intellectual factors in artistic production and his classification of the arts [Productive ideas and know-how] === [[w:Aristotelian physics|Aristotelian physics]] [[w:Theory of impetus|Theory of impetus]] == Part III: Ethics and Politics [Man the Doer] == === 9. The End as the First Principle in Practical thinking and the Use of Means as the Beginning of Action: The End as First in the Order of Intention and Last in the Order of Execution === ''The good as the desirable and the desirable as the good'' * Nicomachean Ethics I.1-2 ''The distinction between ends and means as good desirable for their own sake and goods desirable for the sake of something else.'' * Ethics I.5, I.7, 1.9 ''The ultimate end in practical thinking compared with axioms or self-evident truths in theoretical thinking.'' * Posterior Analytics I.2 === 10. Happiness Conceived as That Which Leaves Nothing to Be Desired and, as so Conceived, the Final or Ultimate End to Be Sought === ''The distinction between living and living well.'' * Politics I.1-2, I.9 ''The conception of happiness as a whole good life, together with various views held by individuals concerning what a good life consists in'' * Ethics I.4-5, I.7-10; X:2, X:6-8 === 11. Distinction Between Real and Apparent Goods, or Between Goods that Ought to Be Desired and Goods That are in Fact Desired, Together with Distinction between Natural and Acquired Desires === === 12. Real Goods === === 13. Moral Virtue and Good Fortune === === 14 Obligations of the Individual === === 15. Role of the State in Abetting or Facilitating the Individual's Pursuit of Happiness === {{Expand section|date=April 2021}} == Part IV: Psychology, Logic, and Theory of Knowledge [Man the Knower] == === 16. The Senses and the Intellect: Perception, Memory, Imagination, and Conceptual Thought === ''Language in relation to thought.'' * Categories I * On Interpretation, 1-2 ''Account of the external senses and of their distinction from the interior senses: the common sense, memory, and imagination.'' * On the Soul II.5-12, III.1-3 * Sense and the Sensible * History of Animals IV.8 ''Distinction between mere sensations and perceptual experience'' * Metaphysics I.1 ''Doctrine that sensations and ideas, taken by themselves or in isolation, are neither true nor false'' * Categories 4 * On Interpretation 1 * On the Soul II.6, III.3, III.6 * Metaphysics IV.5, V.29 === 17. [[w:Immediate_Inference|Immediate Inference]] and [[w:Syllogistic_Reasoning|Syllogistic Reasoning]] === ''The [[w:Law_of_contradiction|law of contradiction]] as an ontological principle and as a rule of thought.'' * On Interpretation 6 * Prior Analytics II.17 * Posterior Analytics I.11 * Metaphysics IV.3-8; IX.5-6 ''The [[w:Square_of_opposition|square of opposition]]: contradictories, contraries, and subcontraries''. Subalterns. * On Interpretation 6, 10 * Categories 10 * Prior Analytics I.2 === 18. Theoretical and Practical Truth === === 19. Theory of Knowledge and Distinction between Knowledge and Right Opinion === {{Expand section|date=April 2021}} == Part V: Difficult Philosophical Questions == {{Expand section|date=April 2021}} == Fair use of a Derivative and Copyrighted Work derived from Primary Sources in the Public Domain == Adler's epilogue is a derivative work of Aristotle's, which are in the public domain. His "Epilogue/second table of contents" is a "selection and arrangement" (and as such may be copyrighted). Reproducing it is necessary to the project to analyzing and evaluating the book, reproducing the TOC is likely to improve rather than hurt sales of the copyrighted book. Furthermore, the original being a derivative of public domain works, it is consistent with the spirit and intent of the author to create further derivative works, and furthermore he has specifically invited his readers and students to "check his exposition" against the originals. For all these reasons and more, I content that this companion course is "fair use" of Adler's "second table of contents" to his book. * [[User:Jaredscribe/WV:TOC is fair use|User:Jaredscribe/WV:Toc is fair use]] And if non-free, and if not allowed under fair use: [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] To host this companion course on Wikiversity would honor this work in particular, would not harm but likely improve sales of his book, and would honor his lifework in general, by its harmony with the manifestos for democratic "public education" in the liberal arts, published by professor Adler and his colleagues in "The [[w:Paideia_Proposal|Paideia Proposal]]" and elsewhere. That there be no doubt, the editors request that Simon & Schuster, Touchstone Book make a [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. This course is being migrated from the wikipedia article [[w:Aristotle_for_Everybody|Aristotle for Everybody]], as it is more appropriate to this project than that. ==Notes== {{reflist}} == Further Reading and Commentary suggested by Wikiversity Professors == {{refbegin}} * {{cite book|title=Aristotle for Everybody: Difficult Thought Made Easy|title-link=Aristotle for Everybody|last=Adler|first=Mortimer|date=1997|publisher=Touchstone|ISBN=0-684-83823-0|location=New York|author-link=w:Mortimer J. Adler|orig-date=1978}} * {{cite book|title=The Basic Works of Aristotle|author=Aristotle|date=1941|publisher=Random House|editor=Richard McKeon|editor-link=w:Richard McKeon|location=New York|author-link=Aristotle}} * {{Cite book|title=Aristotle: A Very Short Introduction|last=Barnes|first=Jonathan|publisher=Oxford University Press|year=2000|isbn=978-0-19-285408-7|author-link=w:Jonathan Barnes}} {{refend}} === Classic and Scholastic Expositors === *[[s:Isagoge (Owen)|Isagoge]] (Introduction) to Aristotle's [[s:Organon_(Owen)/Categories|Categories]] by [[w:Porphyry]] ''not recommended'' but of significant influence in late antiquity through middle ages. *[[s:The_Guide_for_the_Perplexed_(1904)|Guide to the Perplexed]], by [[w:Maimonides|Maimonides]] ''recommended'': its latin translation initiated the [[w:Scholastic|Scholastic]] movement that created the medieval universities. m1vskag72xiktqh84ul69emms72x5kr 0 301480 2690861 2690855 2024-12-08T14:30:40Z Private lecturer (celestial) 2975755 /* Metaphorical language */ The Deluge 2690861 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses virtues like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretation, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. jdorowmr51tp5mz2cw3lmjzz9v9e92n 2690862 2690861 2024-12-08T14:31:02Z Private lecturer (celestial) 2975755 /* The Deluge */ 2690862 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses virtues like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. cj76wyn35g42my2qbj31ycxe8w3t0fi 2690864 2690862 2024-12-08T14:41:09Z Private lecturer (celestial) 2975755 /* What if I feel insecure about my qualification? */ Virtues 2690864 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses virtues like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. 7uj5h6dg02ir1dfpt9ajbm52y4j083e 2690865 2690864 2024-12-08T14:42:49Z Private lecturer (celestial) 2975755 /* Original sin */ [[#Virtues|virtues]] 2690865 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. sw1vk9iy5pkga3vqgea3vdfky0r4nhv 2690866 2690865 2024-12-08T14:50:31Z Private lecturer (celestial) 2975755 /* Image of God */ Will of God 2690866 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|Will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the Will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. rzkhxorl1cm3m3ku1atx8dygtasn2cn 2690868 2690866 2024-12-08T15:03:18Z Private lecturer (celestial) 2975755 /* Will of God */ 2690868 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === [[w:Genesis flood narrative|Genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. 259vpykjur76zn1r4y9urp8gfqhhg0d 2690870 2690868 2024-12-08T15:20:26Z Private lecturer (celestial) 2975755 /* The Deluge */ 2690870 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. j7zwrb89pfgg2yoxi7b8rhyt16ckqow 2690871 2690870 2024-12-08T15:27:38Z Private lecturer (celestial) 2975755 /* Will of God */ university lectures 2690871 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]]. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. 75o053fkny4p3myhrxc8b2c05ac7wda 2690872 2690871 2024-12-08T15:36:18Z Private lecturer (celestial) 2975755 /* The Deluge */ a rather easily foreseeable problem 2690872 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. tuiejyqbqbz8ftfd22qopv87brc6zc2 2690873 2690872 2024-12-08T15:38:04Z Private lecturer (celestial) 2975755 /* The Deluge */ 2690873 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. m7uh1h18oqf713jw43cex1qruveb9n5 2690875 2690873 2024-12-08T15:54:08Z Private lecturer (celestial) 2975755 /* The Deluge */ [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] 2690875 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, equally easily revealed to be agents of evil by heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. 6si26f9bstfwm30ar4jp7f9l93wgjj6 2690879 2690875 2024-12-08T16:24:52Z Private lecturer (celestial) 2975755 /* The Deluge */ 2690879 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, equally easily revealed to be agents of evil by heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. 0zq082jhvdkpnnduty1bdpurun8ns89 2690887 2690879 2024-12-08T18:20:14Z Private lecturer (celestial) 2975755 /* The Great Deluge */ 2690887 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. ocawm0yrgi8kwremmudpemroob13v86 2690892 2690887 2024-12-08T18:49:46Z Private lecturer (celestial) 2975755 /* Education */ Self-fulfilling prophecy against civilisational convergence 2690892 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attention bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as 'collectively intelligent stupidity'. byh7wlh98314gf0t3vs4jocgfx0t9x1 2690893 2690892 2024-12-08T18:52:33Z Private lecturer (celestial) 2975755 /* Self-fulfilling prophecy against civilisational convergence */ [[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]] 2690893 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attention bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. kt1xkifb6zvx6lk9tf88b7168m3eg92 2690985 2690893 2024-12-09T07:56:50Z Private lecturer (celestial) 2975755 /* Self-fulfilling prophecy against civilisational convergence */ 2690985 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. 0exnptyg61467oslzrtjkq69w5snskf 2690995 2690985 2024-12-09T10:49:25Z Private lecturer (celestial) 2975755 {{-}} 2690995 wikitext text/x-wiki [[File:Judicium_Divinum_in_BMPN_2.0.png|thumb|right|577px|Principal workflow]] {{-}} == Metaphorical language == [[File:Funny theory about the ancient kingdom of Edom.png|right|float]] === Evolution vs. creationism === Evolution represents the predator while creationism represents civilization. Obviously evolution favors the predator as the often most intelligent being and therefore the predator is a winner. Thus the metaphorical dispute about evolution vs. creationism should much rather be the topic of whether and how the civilization can dominate the predator sufficiently. Angels are referred to as "created beings", which implies a state of pure civilization (apart from the fact that angels are created beings, while the evolution that created the homo sapiens was both, evolution and creation at the same time, but this is just fact, not metaphor). === Sodom and Gomorrah === The tale of [[w:Sodom and Gomorrah|Sodom and Gomorrah]] tells the story of a city that was apparently bombed, or something very like that. The archfather Abraham negotiates with God that the city should be spared if 10 righteous (starting from 50 righteous) can be found within the city. The metaphor here is that ten percent is a sorry yield rate and that discarding ninety percent of the population as predators is as if asking God to bomb whole cities. Abraham negotiating down from fifty percent to ten percent is, of course, the wrong direction and would make him look bad, but as the archfather of the Jews he lived in an early era that could not have benefitted from good education, because there were no Jews yet. The perspective of the tale is, of course, the biblical message, that [[w:Judaism|Judaism]] (or rather [[w:Yahwism|Yahwism]]) addressed this issue (which it, in fact, does). ==== Social network ==== Easily deduced is the problem of social networks. Lot's wife "looked back to the city" (which was prohibited) and turned into a pillar of salt. Logically there is a social network surrounding any citizen (e.g. Lot) and his wife would be a person who, especially in ancient times, can easily be imagined to be the one to go to the market place and gossip, leading to a social network of people she may be unwilling to give up. If some people go to heaven while others do not this network must be disassembled somewhere. It may seem an unlikely disassembly to take away somebody's wife, but society consists mostly of interrelated families. Logically there is no other point where disassembly can occur, if can merely shift to other families. Thus the message here is that good ethical education is important and the family should hold together and form a sufficiently strong social network and then that disassembly logically cannot happen in one's own family. But why was Lot's wife turned into a pillar of salt? It may not have been her own failure, but strong social ties to predators and thus one is responsible for one's social network. People who are important should have received sufficient ethical education to make disassembly sufficiently unlikely and all other people should be sufficiently irrelevant to make Lot's wife not "look back". This aspect of the tale therefore explains that some people may be admitted (Lot as a nephew of Abraham is admitted), but people close to them may have failed so badly that they have to be excluded (the majority of the city's inhabitants). In the tale the link from one side to the other is necessarily very short and somebody has to lose. Of course one can only speculate about why Lot didn't like his wife enough or why she was better acquainted with other people, but the true meaning is that society consists of families. Lot's family is thus metaphorically an arbitrary family, but in the unlikely situation of being surrounded by the city's inhabitants, who are all doomed. If the network has to break it has to break within a family, consequently it has to break in this family. This being understood, all families should aim not to be in this situation and the perfect society would result. ==== The Sodom and Gomorrah equation ==== The Sodom and Gomorrah equation can be interpretatively gained from the tale. The equation basically says that Jews (the [[w:in-group|in-group]] of the Bible, which can, of course, be extended to include any ethically responsible culture, for instance Christianity, as one of the dominant examples for such an extended in-group) do have ethical mentors, who form a chain of mentors (described by the Archfather() relation), that links them to an angel. The angel here being a metaphor for a human being with an excellent prognosis for going to heaven and becoming "like an angel". Abraham is, of course, in the biblical context not officially referred to as an angel, but he speaks with God, which is meant to convey a similar status ("speaking with God ''like'' an angel"). : &forall; j &in; JEWS &exist; a &in; ANGELS: Archfather (j) = a The necessity for ethical mentoring (or equivalent education) is what the equation describes and the quality of that education may not be arbitrary, but must, so to speak, be certified by an angel, or may otherwise be insufficient. The inhabitants of the city, of course, logically had no chance to have Abraham as the archfather, because when he still was alive he was not able to at the same time be the archfather of Yahwism. What should be easy to deduce is, of course, that the mentoring function archfather() requires too much time, because it requires many generations to become the archfather of a population. Thus a sensible relation would be called archmentor() or archteacher() and create a chain of mentors within the living population. ==== Angels cannot guarantee what they do not control ==== At the same time the tale warns that angels cannot guarantee what they do not control. Abraham, one should assume, would have included Lot's wife personally as a personal acquaintance, but he was not present in the city at the time of destruction. Thus the mentoring chain logically cannot be fully certified by a single person and can still break, if people fail to understand and apply moral culture and ethical standards in their lives, as the people of Sodom and Gomorrah supposedly did. ==== Can a live after death be guaranteed? ==== More usually there is no guarantee that any particular person will enjoy a life after death. The guarantee is more systematically anchored in society itself and thus in the social networks that constitute society, but may be limited by people's moral culture and ethical standards. Consequently there is also no guarantee for a society that it must include persons who will go to heaven. In the tale of Sodom and Gomorrah Lot just leaves the city. Logically he could have done so at any time and then the society of Sodom and Gomorrah would no longer have contained the tiny group of righteous people from his family, thus turning the society of Sodom and Gomorrah into a doomed society without anybody ascending to heaven. ===== Self-fulfilling prophecy ===== Consequently one should strive to be a morally and ethically acceptable person until oneself is satisfied with the result and that should in theory be sufficient motivation to accomplish the goal. Life after death is meant to be a self-fulfilling prophecy and thus the aim to join heaven is meant to be the salvation, but without legalizing arbitrary misconduct, of course, and with increasing ability to act and intelligence comes also increasing responsibility to do so. === Image of God === The [[w:Image of God|Image of God]] is a metaphor with multiple meanings. One meaning is that the [[w:Kingship_and_kingdom_of_God|Kingdom of Heaven]] is not actually a monarchy. Angels do have [[w:free will|free will]], of course; everything else should be unimaginable. The monarchy of heaven is thus rather a democracy, but a democracy with the unimaginable perfection to act in consensus, according to the will of God, thus every voter is a constituent of the group that confirmed or defined the will of the sovereign of heaven. By human standards this could easily be discarded as impossible to achieve, but in heaven this is the goal, because one is civilized and all voters thus strive for the perfect consensus as a cultural dimension. (One is a very cultural dimension up there in heaven.) In theory angels would take the time to educate each other sufficiently until perfection becomes possible, but that is, given the assembled education, wisdom and intelligence, of course, usually not required. ==== Will of God ==== The culture in heaven endorses and requires willingness to negotiate. And what must be negotiable is the logical and responsible [[w:Will of God|will of God]], as determined in the consensus democracy of heaven, which must be limited by ethically and morally possible consensus, because rejecting the consensus obviously cannot be part of the will of God, if God is that sovereign of heaven and consensus is required. Quod erat demonstrandum. A driver towards the [[w:omniscience|omniscience]] of all inhabitants of heaven is that culturally every extended explanation, including university lectures of any scale, are appreciated and accepted, even from a political opponent, because, of course, time is available in any quantity, literally endless. ==== Failure to reach consensus ==== The question if God can move an [[w:Irresistible_force_paradox|immovable object]] is just an invalid question, because immovable objects do not exist. More disconcerting is the issue of problems that do not have perfect solutions. (Another tale tells that Zeus, Lord of the Sky, has been known to have turned such a paradox into [[w:Teumessian_fox|static constellations in heaven]].) Of course heaven can fail to reach consensus, because the perfect choice may not exist. It is easy to construct choices where there is no ideal decision. Given a failure to reach consensus heaven can, as one possible option, agree to disagree and postpone the result until a desirable or required consensus can be reached. Sometimes heaven may act conservatively because of the goal to reach consensus and reluctance to change a previous perfect decision. One could see the Peaceable Kingdom as an example for such a situation: It is the perfect decision to demand of humanity to fulfill human rights as a convergence criterion. Acting conservatively heaven would hesitate to come to a new evaluation of the situation, since the previous perfect consensus decision still seemed quite reasonable. Thus slow progress in the human rights situation may be seen as irrelevant, even though observers might be inclined to see the positive change as an indicator for the final success to tame the predator. ==== Priesthood of all believers ==== The priesthood of all believers is the concept, that all believers do have a natural obligation (like a [[#Lex_naturalis|natural right]], only obligation instead of right) to conduct ethical education and that can easily be deduced to apply, for instance in order to reach consensus or to create ethical [[#Social_network|social networks]] and to be an [[#The_Sodom_and_Gomorrah_equation|ethics mentor]] in order to make people [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|suitable candidates for heaven]]. Thus the obligation exists automatically (is a natural obligation). Quod erat demonstrandum. === The devil === The devil would be a fallen angel communicates a distinction between angel and devil and the devil is no longer an angel. This implies that doing [[w:Good|good]] is no license for doing [[w:Good_and_evil|evil]]. The devil is just a devil, because the virtues, values and goodness of the angel do not compensate the evil of his terror. This is especially true because virtues, values and goodness are the expected standard in heaven, so being good is not exceedingly noteworthy by itself. === Original sin === Original sin means that everybody who is born does have a moral obligation (not actually guilt, of course). A yet somewhat insufficient attempt to describe this moral obligation is the [[w:Declaration of Human Duties and Responsibilities|Declaration of Human Duties and Responsibilities]]. Logically one must possess an obligation to perform certain tasks and duties. For instance all tasks and duties required by the Heaven’s Gate must be performed by citizens without financial motivation, or may (at least metaphorically, following the categorical imperative) not be performed. {{/omitted text}} A more complete version of human duties is easily deduced to include peacekeeping diplomacy, but also cultural mentoring, pacifist education, cultural social networking, integration of immigrants and adolescents, cultural rejection of decadence, cultural rejection of corruption, cultural ethical education and mentoring, cultural community building as an obligation, ethical and psychological qualification and certification and cultural upbringing that endorses [[#Virtues|virtues]] like responsibility, duty, pacifism, educational affinity, discipline, ethics, self-criticism and tolerance. === Love of enemies === One interpretation of [[w:love of enemies|love of enemies]] is the fulfillment of [[#Lex_naturalis|natural rights]] in the [[#The_Peaceable_Kingdom|Peaceable Kingdom]]: Even if somebody is seen as an adversary all his basic rights should be guaranteed. An interpretation of “love of enemies” as natural rights are the [[w:Geneva Conventions|Geneva Conventions]]. Other interpretations include the [[w:right to education|right to education]] in school, if supported by critics of the pupil in question, for instance through mentoring, or fulfillment of basic rights in other countries one may not see as worthy, but grant basic rights to as a matter of principle. === The Great Deluge === The [[w:Genesis flood narrative|genesis flood narrative]] does have multiple interpretations, as usual, but one interpretation is a valid warning about [[w:climate change|climate change]], which certainly constitutes a rather easily foreseeable problem, especially from the omniscient perspective. Significant drivers of climate change are, of course, easily revealed to be agents of evil by omniscient heavenly justice, so climate change can be seen as a very relevant topic for the [[#Is_it_true_that_there_will_be_a_judgment_of_one's_sins?|judgment of one's sins]] in heaven. == Judgment == === Legal standards === A relevant legal standard in heaven is the non-exploitation of the regulatory framework, meaning an intention to explicitly use the regulatory framework as a source of behavior near the lowest common denominator can be punishable. Jeff Bezos, for instance, explicitly once referred to the lowest common denominator as his guiding principle and would thus be punishable under this legislation. The Twelve Apostles do have the slightly humorous, but still serious, additional connotation that ten letters of personal ethics would be required for ethical certification and thus eleven letters would be seen as exploitation of the regulatory framework, making twelve the minimum number of ethics mentors required for certification. ==== Nulla poena sine lege ==== As a consequence nulla poena sine lege (no penalty without law) would also not be applied as strictly in heaven, meaning the regulatory framework is allowed to differ from the expectation, especially for juridical persons (who should have been striving for higher goals than the lowest common denominator to barely be within legal requirements) and especially as an option for the court to either apply or not apply older or newer legislation to a case. On the other hand the very ancient legislation of heaven, of course, does not change very much anyway and the judges are, of course, omniscient, meaning they will not misapply this opportunity, but find the perfect judgement. ==== The Twelve Apostles ==== The Twelve Apostles represent the social network of Jesus as a duality, the state of the social network being a variable depending on the (existence or non-existence of) culture. From inside Christianity the culture would certainly be Christian, but otherwise it would be undefined. {{/omitted text}} Thus the importance of the social network is emphasized and Jesus as another “angel” would “certify” the social network of the Twelve Apostles, but the Twelve Apostles would also mutually “certify” the ethical standards (teachings) of Jesus, thus create a mutually certified ethical social network. In the absence of any certification there is, of course, no strict requirement on Earth. Ten would be the sensible requirement, that is easily invented and understood. Non-exploitation of the regulatory framework is easily applied to this new regulation, even if not strictly specified to apply, so this would more be an interpretation by superiors, but not strictly required. Alternatively one could also observe that a minimum fulfillment would show that apparently the topic had not been interesting enough. Consequently, because – wanting to be prepared – one should logically want to fulfill this requirement for most of one’s lifetime and one would have at least ten to twelve ethics mentors from adolescence, but later in life would permanently seek to gain new ethics mentors and new certifications, especially when rising in rank oneself, because mentors from adolescence can easily be perceived as very insufficient later in life and especially by superiors. Pensioners could again see a need to improve this network, because their perspective would more focus on a future in heaven and thus provide new motivation. 120 cardinals form a papal conclave, which would, of course, be over-fulfillment, but understandably serve the '''very''' purpose. The Twelve Apostles, being both young adults or adults, would also be two groups at once, thus the “earlier 12” or the “later 12”. Jesus apparently also would have had Twelve Apostles at about the age of thirty, which would be an age where ascension in society could motivate exactly the behavior to form new relationships with the second group of mentors. One wouldn’t expect a man at that age to die at all, but – wanting to be prepared – one would maintain the perspective and resulting motivation and thus continue to build a social network of ethics mentors. The apostles are later mentioned as visitors in Rome, Athens and other cities and as old men, which would make this a reference to the third group of ethics mentors, one would gather as a pensioner. Also the network apparently would in that era count as “worldwide”, so pensioners are presented as having the opportunity to extend their network to, at least, other cities, but in effect contributing to worldwide networking. ==== Ignorantia legis non excusat ==== Also the Heaven’s Gate does, logically, not strictly apply ignorantia legis non excusat (ignorance of the law is no excuse), because, quite clearly, ignorance should have a (very limited) power to excuse at the Heaven’s Gate. ==== Lex naturalis ==== Lex naturalis (natural law) is seen as to dominate over subordinate legislation and the resulting problem of financial assets is (while not being relevant anyway) lessened by founding the financial systems in contractual law, meaning use of any financial system first requires a founding contract and there is no national financial system to compete with that. The advantage is that, as in the Jewish culture, all contracts are subject to the cultural (e.g. rabbinical, beth din) courts required by the cultural social contract and are therefore necessarily in agreement with the intended culture. Jesus supposedly responded to a question about taxation with the well-known quote “Render therefore unto Caesar what is Caesar's; and to God what is God's.” (Matthew 22:21). A son of God would {{/omitted text}} and consequently in theory utilize multiple financial systems, but be himself, as a citizen of utopia (a “holy man”, mankind is holy – all basic rights fulfilled), be above the need for finance. ===== Son of God ===== Holiness of mankind would be another reference to human rights as the [[#The_Peaceable_Kingdom|convergence criteria]]: The holy man is the Son of God, has a “holy” certification and can then ascend to heaven. The Son of God metaphor would also carry the meaning that the social network on Earth would somehow have to undergo a kind of tunnel effect to suddenly contain members of the social network in heaven. The magic of that tunnel effect would be adoption. And adoption could be adoption of a child or adoption of a culture and ethical standards, both of which have a potentially beneficial effect. Adoption of a young adult on a university would, for instance, naturally occur by a doctoral advisor (German Doktorvater means “doctor father”) and could, of course, be easily envisioned to occur through an omniscient celestial doctoral advisor. === Is it true that there will be a judgment of one's sins? === That is definitely true and because angels watch everything humans do the judgment starts immediately with the sin, usually not much later. Mankind does, however, not have a reliable book of law that would detail the actual laws of heaven. All works that try to describe heavenly law were written by humans and contain cultural bias, human opinion and moral standards considered adequate at the time of writing. They may, of course, also contain an unknown amount of fact and/or metaphorical language originating in heaven. The educated reader may be able to distinguish the different types of content. As tourists people often travel to foreign countries without first learning all their laws. It is thus not really unusual not to be aware of the legislation of a state. As a rule of thumb any legislation can be approximated with the [[w:categorical imperative|categorical imperative]], especially heavenly law favors the categorical imperative and resulting moral culture and ethical standards. === The Peaceable Kingdom === The [[w:Peaceable Kingdom (theology)|Peaceable Kingdom]] is a future society that is supposed to precede the [[#Image of God|Kingdom of Heaven]]. What this actually means is that the predator (the homo sapiens is a predator) must be tamed and that people do have [[#Lex_naturalis|natural rights]], which must be guaranteed. The Peaceable Kingdom is thus neither more nor less than a future state of society in which natural rights are sufficiently guaranteed. This is a necessary, but not a sufficient convergence criterion for the Kingdom of Heaven. The Kingdom of Heaven will require even higher standards and human rights that do not even exist as human rights today. The land [[w:Canaan|Canaan]] is associated with the Biblical [[w:Promised Land|Promised Land]], which can be reinterpreted as a promised territory in which migrants find refuge and this then would metaphorically and applying the [[w:categorical imperative|categorical imperative]] include heaven as a refuge for humanity for a live after death. According to the categorical imperative, of course, one should strive to provide refuge to migrants, especially during climate change, who may otherwise not survive in their state of origin, and thus in part satisfy the convergence criterion Peaceable Kingdom. === Duality of personal future and the future of mankind === The duality of one's personal future and the future or mankind is meant to convey that one should aim for a future of mankind that is desirable. Climate change, for instance, makes it perfectly clear that an imaginable future of humanity is a catastrophic disaster. One should, of course, choose not to be the cause of a catastrophic disaster or the all-knowing judge in heaven would have to regard that as a very serious misconduct. As a rule of thumb it makes sense to aim for a future of humanity in heaven that can actually occur, or one will not be able to enjoy it. This should be seen to include the Peaceable Kingdom as a convergence criterion: If you choose to stay divergent, applying the categorical imperative, there would as a result be no future in which you could ascend to heaven. That is, of course, not actually true. Others may create the future without your help, but the judge in heaven may object to your presence in heaven, depending on your personal misconduct, thus making the duality come true. === Is education important for the judgment or just good conduct? === Education is a very positive cultural trait, but not strictly necessary. What is urgently required is ethical education that is sufficient so that the individual has a positive prognosis to become a good citizen of heaven. Strict adherance to a sufficient religion would thus constitute a good standard to receive such a positive prognosis, but heaven aims to make perfect decisions, so that should better be a credible judgment. For instance acceptance of God in heaven as the undisputed sovereign and strict pacifism are very positive cultural traits, even lacking higher education, that could otherwise be seen as a qualifying criterion. Heaven is, however, also very selective about which higher education that would be and consequently one is definitely well advised to consider the constitution of heaven as God-given and pacifism as a self-evident necessity. Of course the inhabitants of heaven enjoy natural rights and among them are the rights to freedom of thought and freedom of speech, but the constitution of heaven should be seen as immutable and thus the free will to endorse the constitution that guarantees these rights is also a very positive cultural trait, thus heaven would be, so to speak, a monarchy (as opposed to anarchy). === What if I feel insecure about my qualification? === People can join heaven as a result of their social network requesting their presence, but only if that is permitted by the judge of heaven and subordinate authorities. There may also be unexpected problems to this approach that are not well-suited for public debate, so the recommended practice is to form an adequate social network in advance, preferably with the explicit purpose of getting one into heaven. Since the society in heaven has a tendency to become more educated over time the likelihood of a good teacher from your personal social network becoming available as mentor rises constantly. What is beneficial is a good social network, that engages in mentoring, and acceptance for people you know as mentors, that may be willing to help, on your side. Any Christian priest could be seen to fulfill that requirement for his parish, which is because that is the God-given intended function. That is, of course, again no license for sever misconduct, because the judge in heaven can object permanently. The [[Ethics/Life_after_death#The_devil|devil]] is such a theoretical terrorist, who can not be allowed to enter heaven, or would have to be expelled by force. The ability to enter heaven without permission is, however, a rather theoretical thing. Angels would be able to try, but they don't do that. In an existential sense the devil is not just a theory and does exist, but he may also be encountered in actions by persons who fail to employ sufficient ethical standards and as a result act as if instructed by such an agent of evil. Heaven refers to the latter as 'collectively intelligent stupidity' or just stupidity, because one should be able to deduce that it may cause incalculable problems for one's personal future in heaven, which should logically enjoy the highest priority or be among the highest priorities. ==== Virtues ==== “I am superior to the other” is an attitude that may emerge from various cognitive biases. There is an interesting observation to be made: Allowing others to be good enough, but questioning oneself whether one is good enough, even if the opposite perception arises, is a sensible cultural trait. Obviously one can benefit from self-criticism for self-improvement and one can never be sure to qualify against the not well-defined requirements of heaven, so the sensible attitude is to strive for a higher standard oneself, at least until one feels sufficiently confident about one’s own qualification, even against unknown requirements. Allowing the other to be good enough to qualify, on the other hand, means others may be worthy of attention and support, possibly resulting in mentoring, and to avoid conflict that could be prejudicial, which is very clearly a beneficial situation for society. People may also feel very differing inclination to strive for higher standards. Self-criticism and tolerance, despite a possibly opposite perception, allow individuals to be driven by a higher standard and thus to take on important roles in society, where behavior near the lowest common denominator is no alternative. Consequently, self-criticism and tolerance are also relevant virtues. Quod erat demonstrandum. == Science == === Will science allow us to gain all the magic of heaven and do without it? === No, it won't, but that is a rather complicated analysis and you are, of course, allowed to believe in science. === Is physical entry into the otherworld possible? === Entry into the [[w:otherworld|otherworld]] is not physically possible. If it were possible normal matter (water) would become exotic matter (wine), organic chemistry and especially protein folding would break down and containers would cease to contain their content. Trivially these conditions would be unhealthy for the traveler, but this is a theoretical problem, because matter does not travel to the otherworld at all. What can enter the otherworld is only the soul, which is pure energy, light and information. It can enter the otherworld because it does not physically exist and (notice the change of interpretation) the soul in its non-existence is about virtues, values and goodness. It, however, has no need to travel, because it resides already in the otherworld. === Can the soul come back to this world? === There are multiple issues that are not well-suited for public debate, especially not, given the different interpretations of different religions, but in theory this is possible and if an angel would be sitting in a barrack somewhere in Africa and waiting for his natural rights to be acknowledged you wouldn't be able to tell the difference. He might, of course, leave once his natural rights had been granted and could, for instance, simultaneously reside in the otherworld and sit in parliament as a special rapporteur on human rights. This is very definitely possible, but not very likely, rather an adequate metaphor for the possibility and the goal to fulfill human rights. === Is the soul immortal and eternal? === There are different ways to see this. What is most important is that the soul should be seen as an integral part of the human being from somewhere between conception and birth on. Whether it exists before conception or not is, again, not well-suited for public debate and a somewhat academic question: Yes and No. Only this way, from birth on, the soul can grant the most perfect immortality that can be conferred. It is certainly eternal in the sense that it does not have a limited life time. == Education == === A proposal for better education === Useful appears to be the goal to make pupils envision their own path to heaven, for instance as a repeating home work, refining that goal every year during middle school and high school and freely developing and researching their own perspective on the topic. Developing one’s own perspective with independent and creative thought is good on the one hand, but on the other hand it is actually not reliable enough and thus one would complement that with cultural education that defines cultural limitations and certification, for instance through ethics mentors (like, metaphorically, [[#The_Twelve_Apostles|the apostles]]) or equivalent education. Freedom of thought appears necessary and desirable, but a certain limitation of the resulting culture also appears to be indispensable, just as the logical and responsible Will of God must be limited by [[#Failure_to_reach_consensus|ethically and morally possible consensus decisions in heaven]]. A potential problem of an increased believe in an afterlife can, however, also increase the risk of teenager suicide, so one would logically restrict this pedagogy to teenagers where no such risk is allowed to occur. Unfortunately this would mean that in general this pedagogy cannot be recommended to arbitrary families. === Self-fulfilling prophecy against civilisational convergence === This negative prophecy would benefit from cognitive biases like [[w:choice-supportive bias|choice-supportive bias]], [[w:hyperbolic discounting|hyperbolic discounting]], [[w:present bias|present bias]] and [[w:attentional bias|attentional bias]]. Due to attentional bias for instance, theists are known to confirm that God answers prayers. More relevant would be the observation that theists, due to attentional bias, have a stronger tendency to believe in and prepare for an afterlife, while atheists are less likely to do so. It follows that more attention to the topic is psychologically advantageous in order to maintain (to avoid the word belief) the sensible strategy. Choice-supportive bias also supports the decision of atheists not to pay attention to religion and the afterlife, or, at least, the sensible strategy and that in favor of temporal closer rewards (hyperbolic discounting, present bias), but thus contributing to the self-fulfilling prophecy against civilisational convergence. But since [[w:Pascal's wager|Pascal's wager]] correctly described the sensible choice this could be seen as '[[#What_if_I_feel_insecure_about_my_qualification?|collectively intelligent stupidity]]'. f3wi0j851zoun42dhlje1onh7eq6e16 0 305362 2690904 2690724 2024-12-08T21:35:00Z Dc.samizdat 2856930 /* Reflections */ simplify misleading description 2690904 wikitext text/x-wiki {{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]]<br>(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}<br>r{3,3,4} = <math>\left\{\begin{array}{l}3\\3,4\end{array}\right\}</math><br>{3^1,1,1} = <math>\left\{\begin{array}{l}3\\3\\3\end{array}\right\}</math> | CD={{Coxeter–Dynkin diagram|node_1|3|node|4|node|3|node}}<br>{{Coxeter–Dynkin diagram|node|3|node_1|3|node|4|node}} or {{Coxeter–Dynkin diagram|node_1|split1|nodes|4a|nodea}}<br>{{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)|F₄]], [3,4,3], order 1152<br>B₄, [4,3,3], order 384<br>D₄, [3^1,1,1], 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 '''C₂₄''', or the '''icositetrachoron''',{{Sfn|Johnson|2018|p=249|loc=11.5}} '''octaplex''' (short for "octahedral complex"), '''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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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="wikitable floatright" |+ [[W:Orthographic projection|Orthogonal projection]]s of the 24-cell |- style="text-align:center;" ![[W:Coxeter plane|Coxeter plane]] !colspan=2|F₄ |- style="text-align:center;" !Graph |colspan=2|[[File:24-cell t0_F4.svg|100px]] |- style="text-align:center;" ![[W:Dihedral symmetry|Dihedral symmetry]] |colspan=2|[12] |- style="text-align:center;" !Coxeter plane !B₃ / A₂ (a) !B₃ / A₂ (b) |- style="text-align:center;" !Graph |[[File:24-cell t0_B3.svg|100px]] |[[File:24-cell t3_B3.svg|100px]] |- style="text-align:center;" !Dihedral symmetry |[6] |[6] |- style="text-align:center;" !Coxeter plane !B₄ !B₂ / A₃ |- style="text-align:center;" !Graph |[[File:24-cell t0_B4.svg|100px]] |[[File:24-cell t0_B2.svg|100px]] |- style="text-align:center;" !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 = 24².{{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=; "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.)".}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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. <math display="block">\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math> 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)|F₄]] group, as shown in this F₄ Coxeter plane projection]] The 24 root vectors of the [[W:D4 (root system)|D₄ 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=: "... 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."}} 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 B₄ and C₄ 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)|F₄]].{{Sfn|van Ittersum|2020|loc=§4.2.5|p=78}} The 24 vertices of the original 24-cell form a root system of type D₄; 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 F₄, which is generated by [[W:Reflection (mathematics)|reflections]] through the hyperplanes orthogonal to the F₄ 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 F₄ [[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 D₄ root lattice is the [[W:Dual lattice|dual]] of the F₄ 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)|D₄]] 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 D₄ lattice points (Hurwitz quaternions with even norm squared) while the vertices are at the F₄ 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'''⁴. 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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]] 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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]] 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 "[[W:North Pole|North Pole]]". Eight great circle meridians (two cells long) radiate out in 3 dimensions, converging at the 3rd "[[W:South Pole|South Pole]]" cell. This skeleton accounts for 18 of the 24 cells (2&nbsp;+&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 "equatorial" 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="wikitable" |- ! Layer # ! Number of Cells ! Description ! Colatitude ! Region |- | style="text-align: center" | 1 | style="text-align: center" | 1 cell | North Pole | style="text-align: center" | 0° | rowspan="2" | Northern Hemisphere |- | style="text-align: center" | 2 | style="text-align: center" | 8 cells | First layer of meridian cells | style="text-align: center" | 60° |- | style="text-align: center" | 3 | style="text-align: center" | 6 cells | Non-meridian / interstitial | style="text-align: center" | 90° | style="text-align: center" |Equator |- | style="text-align: center" | 4 | style="text-align: center" | 8 cells | Second layer of meridian cells | style="text-align: center" | 120° | rowspan="2" | Southern Hemisphere |- | style="text-align: center" | 5 | style="text-align: center" | 1 cell | South Pole | style="text-align: center" | 180° |- ! Total ! 24 cells ! colspan="3" | |} [[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 "vertical", 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]]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="wikitable" width=660 !colspan=3|Cell-first perspective projection |- valign=top |[[Image:24cell-perspective-cell-first-01.png|220px]]<BR>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]]<BR>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]]<BR>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 "equator" of the 24-cell, and bridge the two sets of cells. |} {| class="wikitable" width=440 |[[Image:24cell section anim.gif|220px]]<br>Animated cross-section of 24-cell |- |colspan=2 valign=top|[[Image:3D stereoscopic projection icositetrachoron.PNG|450px]]<br>A [[W:Stereoscopy|stereoscopic]] 3D projection of an icositetrachoron (24-cell). |- |colspan=3|[[File:Cell24Construction.ogv|450px]]<br>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 B₄ or [3,3,4] symmetry drawn bicolored with 8 and 16 [[W:Octahedron|octahedral]] cells. Lastly it can be constructed from D₄ or [3^1,1,1] symmetry, and drawn tricolored with 8 octahedra each.<!-- 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 */ --> {| class="wikitable collapsible collapsed" !colspan=12| Three [[W:Net (polytope)|nets]] of the ''24-cell'' with cells colored by D₄, B₄, and F₄ symmetry |- ![[W:Rectified demitesseract|Rectified demitesseract]] ![[W:Rectified demitesseract|Rectified 16-cell]] !Regular 24-cell |- !D₄, [3^1,1,1], order 192 !B₄, [3,3,4], order 384 !F₄, [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]]'''<br>(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]] ₄{3}₄, {{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 ₄[3]₄, order 96.{{Sfn|Coxeter|1991|p=}} The regular complex polytope ₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} or {{Coxeter–Dynkin diagram|node_h|8|3node}}, in <math>\mathbb{C}^2</math> has a real representation as a 24-cell in 4-dimensional space. ₃{4}₃ has 24 vertices, and 24 3-edges. Its symmetry is ₃[4]₃, 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}} !₄{3}₄, {{Coxeter–Dynkin diagram|4node_1|3|4node}} !₃{4}₃, {{Coxeter–Dynkin diagram|3node_1|4|3node}} |- !Symmetry ![3,4,3], {{Coxeter–Dynkin diagram|node|3|node|4|node|3|node}}, order 1152 !₄[3]₄, {{Coxeter–Dynkin diagram|4node|3|4node}}, order 96 !₃[4]₃, {{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]]<BR>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]]<BR>₄{3}₄, {{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]]<BR>₃{4}₃ 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 "[[W:Regular icosahedron#Uniform colorings and subsymmetries|snub octahedron]]." 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}} * {{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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1968 | title=The Beauty of Geometry: Twelve Essays | publisher=Dover | place=New York | edition=2nd }} * {{Cite journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. Applic. | volume=17 | issue=1–3 | pages=59–71 | doi=10.1016/0898-1221(89)90148-X | doi-access=free }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1970 | title=Twisted Honeycombs | place=Providence, Rhode Island | journal=Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics | publisher=American Mathematical Society | volume=4 }} * {{Cite journal|last=Stillwell|first=John|date=January 2001|title=The Story of the 120-Cell|url=https://www.ams.org/notices/200101/fea-stillwell.pdf|journal=Notices of the AMS|volume=48|issue=1|pages=17–25}} *{{Citation | last=Johnson | first=Norman | author-link=W:Norman Johnson (mathematician) | title=Geometries and Transformations | year=2018 | place=Cambridge | publisher=Cambridge University Press | isbn=978-1-107-10340-5 | url=https://www.cambridge.org/core/books/geometries-and-transformations/94D1016D7AC64037B39440729CE815AB}} * {{Citation | last=Johnson | first=Norman | author-link=W:Norman Johnson (mathematician) | year=1991 | title=Uniform Polytopes | edition=Manuscript }} * {{Citation | last=Johnson | first=Norman | author-link=W:Norman Johnson (mathematician) | year=1966 | title=The Theory of Uniform Polytopes and Honeycombs | edition=Ph.D. }} * {{cite book|last=Ghyka|first=Matila|title=The Geometry of Art and Life|date=1977|place=New York|publisher=Dover Publications|isbn=978-0-486-23542-4}} * {{cite book|last=Banchoff|first=Thomas F.|chapter=Torus Decompostions of Regular Polytopes in 4-space|date=2013|title=Shaping Space|url=https://archive.org/details/shapingspaceexpl00sene|url-access=limited|pages=[https://archive.org/details/shapingspaceexpl00sene/page/n249 257]–266|editor-last=Senechal|editor-first=Marjorie|publisher=Springer New York|doi=10.1007/978-0-387-92714-5_20|isbn=978-0-387-92713-8}} * {{Cite arXiv | eprint=1903.06971 | last=Copher | first=Jessica | year=2019 | title=Sums and Products of Regular Polytopes' Squared Chord Lengths | class=math.MG }} * {{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]] r5qbxv696953ya20dukazyx3xbmmd88 0 307507 2690856 2689104 2024-12-08T13:26:27Z Sylvain Ribault 2127778 2690856 wikitext text/x-wiki ''A graduate course of conformel field theory (CFT), with 18 hours lectures and 18 hours tutorials.'' We sketch the main motivations of CFT, including its applications to statistical physics, high-energy physics, and quantum gravity. We introduce CFT in the bootstrap approach, an axiomatic approach that starts from symmetry and consistency conditions for quantum fields, and deduces crossing symmetry equations for correlation functions. For most of the course, we specialize to 2 dimensions, where the existence of infinitely many conformal transformations leads to exact solutions of a number of nontrivial CFTs. We study the relevant technical constructions, from the Virasoro algebra to conformal blocks. Solving crossing symmetry and other constraints or assumptions, we obtain CFTs such as minimal models, Liouville theory and loop models. We also introduce CFTs that have extra symmetries beyond conformal symmetry, such as free bosons and Wess-Zumino-Witten models. == Prerequisites == === Mathematical === {{main|Mathematical prerequisites for 2d CFT}} * Complex analysis: contour integrals of complex analytic functions on <math>\mathbb{C}</math>. * Lie algebras and their representations. === Physical === * Notions of quantum mechanics and quantum field theory. == Course topics == === Conformal symmetry === {{main|Conformal symmetry, its motivations, its applications}} * Scale invariance and conformal invariance. * Fixed points of the renormalization group. * Applications to statistical physics, high-energy physics, quantum gravity. === The bootstrap method === * Bootstrap vs Lagrangian. * Conformal symmetry in d dimensions. * Crossing symmetry. * Unitarity. === Conformal invariance in 2d === * Virasoro algebra and its representations. * Fields, energy-momentum tensor. * OPEs, fusion rules. Example: minimal models. * Correlation functions. Single-valuedness. Example: loop models. * Conformal blocks. === Analytic bootstrap in 2d === * Belavin-Polyakov-Zamolodchikov differential equations. * Shift equations for structure constants. * Double Gamma function and solutions of shift equations. Example: Liouville theory. === Additional symmetries === * Global vs local conformal symmetry. * Discrete symmetries: Ising and Potts models. * Chiral algebra: affine Lie algebras, W-algebras. * Interchiral symmetry. === CFTs based on affine Lie algebras === * Free bosons. * WZW models. == Other topics == These topics are a priori not covered in detail, but they could be mentioned, or be the subjects of student projects: * Boundary CFT, defects. CFT on the cylinder. * Entanglement entropy. * Modular invariance. * Numerical bootstrap. * Coulomb gas approach. * Fusing matrix (= fusion kernel). * Conformal perturbation theory. * Renormalization group flows between CFTs. == Relevant Wikipedia articles == The following Wikipedia articles are particularly relevant to this course. Consulting them can be helpful for seeing the relations of CFT with other subjects, and for finding relevant references. Moreover, student projects may involve criticizing these articles and improving them (see [[w:User:Sylvain_Ribault/YRIS2019|tutorial]]). * [[w: Conformal field theory]]. * [[w: Two-dimensional conformal field theory]]. * [[w: Virasoro algebra]]. * [[w: Minimal model (physics)]]. * [[w: Liouville theory]]. == Resources == (E) = texts with exercises, (ES) = texts with exercises and their solutions. === CFT and bootstrap in general dimensions === * '''EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions'''<ref name="ryc16"/>, by Slava Rychkov, 68 pages: an introduction to CFT that starts with a discussion of the history and ideas, and provides a guide to some of the relevant literature. * '''The Conformal Bootstrap: Theory, Numerical Techniques, and Applications'''<ref name="prv18"/>, by David Poland, Slava Rychkov and Alessandro Vichi, 81 pages: a review article that has much to say on the applications to 3d CFTs. === 2d CFT in the bootstrap approach === * (ES) '''Minimal lectures on two-dimensional conformal field theory'''<ref name="rib16"/>, by Sylvain Ribault, 37 pages: a concise introduction to 2d CFT in the bootstrap approach. * (E) '''Conformal field theory on the plane'''<ref name="rib14"/>, by Sylvain Ribault, 145 pages: an introduction to 2d CFT in the bootstrap approach, including a chapter on affine symmetry. * '''Exactly solvable conformal field theories'''<ref name="rib24"/>, by Sylvain Ribault, 85 pages: an introduction to 2d CFT with an emphasis on exact solvability and on loop models. === 2d CFT from other points of view === * '''Applied Conformal Field Theory'''<ref name="gin91"/>, by Paul Ginsparg, 178 pages: an early review that can still be useful, in particular for its treatment of free fermions and bosons, orbifolds thereof, and CFT on a torus. * '''Conformal Field Theory and Statistical Mechanics'''<ref name="car08"/>, by John Cardy, 37 pages: a concise introduction to 2d CFT from the point of view of statistical mechanics. * (E) '''Conformal Field Theory for 2d Statistical Mechanics'''<ref name="ei23"/>, by Benoît Estienne and Yacine Ikhlef, 150 pages: a course that insists on statistical physics motivations and applications. === Wider horizons === * (E) '''Scaling and Renormalization in Statistical Physics'''<ref name="car96"/>, by John Cardy, 238 pages: an excellent text for understanding the role of CFT in statistical physics, although CFT is not its main subject. * (E) '''Conformal Field Theory'''<ref name="fms97/>, by Philippe di Francesco, Pierre Mathieu and David Sénéchal, 890 pages: the Big Yellow Book on CFT, mostly in 2d, with an in-depth treatment of minimal models and Wess-Zumino-Witten models. == References == {{Reflist|refs= <ref name="prv18">{{cite journal | last=Poland | first=David | last2=Rychkov | first2=Slava | last3=Vichi | first3=Alessandro | title=The conformal bootstrap: Theory, numerical techniques, and applications | journal=Reviews of Modern Physics | publisher=American Physical Society | volume=91 | issue=1 | date=2019-01-11 | issn=0034-6861 | doi=10.1103/revmodphys.91.015002 | doi-access=free | url=https://arxiv.org/abs/1805.04405}}</ref> <ref name="car08">{{cite web | last=Cardy | first=John | title=Conformal Field Theory and Statistical Mechanics | website=arXiv.org | date=2008-07-22 | url=https://arxiv.org/abs/0807.3472 | access-date=2024-11-12}}</ref> <ref name="gin91">{{cite web | last=Ginsparg | first=Paul | title=Applied Conformal Field Theory | website=arXiv.org | date=1988 | url=https://arxiv.org/abs/hep-th/9108028 | access-date=2024-09-03}}</ref> <ref name="ei23">{{cite web | last=Estienne| first=Benoît|last2 = Ikhlef |first2 = Yacine | title=Conformal Field Theory for 2d Statistical Mechanics | date=2023 | url=https://www.lpthe.jussieu.fr/~ikhlef/CFT.pdf | access-date=2024-08-31}}</ref> <ref name="car96">{{cite book | last=Cardy | first=John | title=Scaling and Renormalization in Statistical Physics | publisher=Cambridge University Press | date=1996 | isbn=978-0-521-49959-0 | doi=10.1017/cbo9781316036440 | page=}}</ref> <ref name="rib24">{{cite web | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | website=arXiv.org | date=2024-11-26 | url=https://arxiv.org/abs/2411.17262 | access-date=2024-11-27}}</ref> <ref name="fms97>{{cite book | last=di Francesco | first=Philippe | last2=Mathieu | first2=Pierre | last3=Sénéchal | first3=David | title=Conformal Field Theory | publisher=Springer | date=1997 | isbn=0-387-94785-X | page=}}</ref> <ref name="rib14">{{cite web | last=Ribault | first=Sylvain | title=Conformal field theory on the plane | website=arXiv.org | date=2014 | url=https://arxiv.org/abs/1406.4290 | access-date=2024-08-31}}</ref> <ref name="rib16">{{cite journal | last=Ribault | first=Sylvain | title=Minimal lectures on two-dimensional conformal field theory | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | date=2018 | url = https://arxiv.org/abs/1609.09523| doi=10.21468/scipostphyslectnotes.1 | doi-access=free | page=}}</ref> <ref name="ryc16">{{cite journal | last= Rychkov|first=Slava |date=2016 |title= EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions| journal=SpringerBriefs in Physics |doi=10.1007/978-3-319-43626-5|arxiv=1601.05000|url = https://arxiv.org/abs/1601.05000 |isbn=978-3-319-43625-8 |s2cid=119192484 }}</ref> }} [[Category:Conformal field theory]] [[Category:CFT course]] q5hccg84pc9om0e1dn618t43dsmqd4y 2690857 2690856 2024-12-08T13:26:39Z Sylvain Ribault 2127778 2690857 wikitext text/x-wiki ''A graduate course on conformel field theory (CFT), with 18 hours lectures and 18 hours tutorials.'' We sketch the main motivations of CFT, including its applications to statistical physics, high-energy physics, and quantum gravity. We introduce CFT in the bootstrap approach, an axiomatic approach that starts from symmetry and consistency conditions for quantum fields, and deduces crossing symmetry equations for correlation functions. For most of the course, we specialize to 2 dimensions, where the existence of infinitely many conformal transformations leads to exact solutions of a number of nontrivial CFTs. We study the relevant technical constructions, from the Virasoro algebra to conformal blocks. Solving crossing symmetry and other constraints or assumptions, we obtain CFTs such as minimal models, Liouville theory and loop models. We also introduce CFTs that have extra symmetries beyond conformal symmetry, such as free bosons and Wess-Zumino-Witten models. == Prerequisites == === Mathematical === {{main|Mathematical prerequisites for 2d CFT}} * Complex analysis: contour integrals of complex analytic functions on <math>\mathbb{C}</math>. * Lie algebras and their representations. === Physical === * Notions of quantum mechanics and quantum field theory. == Course topics == === Conformal symmetry === {{main|Conformal symmetry, its motivations, its applications}} * Scale invariance and conformal invariance. * Fixed points of the renormalization group. * Applications to statistical physics, high-energy physics, quantum gravity. === The bootstrap method === * Bootstrap vs Lagrangian. * Conformal symmetry in d dimensions. * Crossing symmetry. * Unitarity. === Conformal invariance in 2d === * Virasoro algebra and its representations. * Fields, energy-momentum tensor. * OPEs, fusion rules. Example: minimal models. * Correlation functions. Single-valuedness. Example: loop models. * Conformal blocks. === Analytic bootstrap in 2d === * Belavin-Polyakov-Zamolodchikov differential equations. * Shift equations for structure constants. * Double Gamma function and solutions of shift equations. Example: Liouville theory. === Additional symmetries === * Global vs local conformal symmetry. * Discrete symmetries: Ising and Potts models. * Chiral algebra: affine Lie algebras, W-algebras. * Interchiral symmetry. === CFTs based on affine Lie algebras === * Free bosons. * WZW models. == Other topics == These topics are a priori not covered in detail, but they could be mentioned, or be the subjects of student projects: * Boundary CFT, defects. CFT on the cylinder. * Entanglement entropy. * Modular invariance. * Numerical bootstrap. * Coulomb gas approach. * Fusing matrix (= fusion kernel). * Conformal perturbation theory. * Renormalization group flows between CFTs. == Relevant Wikipedia articles == The following Wikipedia articles are particularly relevant to this course. Consulting them can be helpful for seeing the relations of CFT with other subjects, and for finding relevant references. Moreover, student projects may involve criticizing these articles and improving them (see [[w:User:Sylvain_Ribault/YRIS2019|tutorial]]). * [[w: Conformal field theory]]. * [[w: Two-dimensional conformal field theory]]. * [[w: Virasoro algebra]]. * [[w: Minimal model (physics)]]. * [[w: Liouville theory]]. == Resources == (E) = texts with exercises, (ES) = texts with exercises and their solutions. === CFT and bootstrap in general dimensions === * '''EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions'''<ref name="ryc16"/>, by Slava Rychkov, 68 pages: an introduction to CFT that starts with a discussion of the history and ideas, and provides a guide to some of the relevant literature. * '''The Conformal Bootstrap: Theory, Numerical Techniques, and Applications'''<ref name="prv18"/>, by David Poland, Slava Rychkov and Alessandro Vichi, 81 pages: a review article that has much to say on the applications to 3d CFTs. === 2d CFT in the bootstrap approach === * (ES) '''Minimal lectures on two-dimensional conformal field theory'''<ref name="rib16"/>, by Sylvain Ribault, 37 pages: a concise introduction to 2d CFT in the bootstrap approach. * (E) '''Conformal field theory on the plane'''<ref name="rib14"/>, by Sylvain Ribault, 145 pages: an introduction to 2d CFT in the bootstrap approach, including a chapter on affine symmetry. * '''Exactly solvable conformal field theories'''<ref name="rib24"/>, by Sylvain Ribault, 85 pages: an introduction to 2d CFT with an emphasis on exact solvability and on loop models. === 2d CFT from other points of view === * '''Applied Conformal Field Theory'''<ref name="gin91"/>, by Paul Ginsparg, 178 pages: an early review that can still be useful, in particular for its treatment of free fermions and bosons, orbifolds thereof, and CFT on a torus. * '''Conformal Field Theory and Statistical Mechanics'''<ref name="car08"/>, by John Cardy, 37 pages: a concise introduction to 2d CFT from the point of view of statistical mechanics. * (E) '''Conformal Field Theory for 2d Statistical Mechanics'''<ref name="ei23"/>, by Benoît Estienne and Yacine Ikhlef, 150 pages: a course that insists on statistical physics motivations and applications. === Wider horizons === * (E) '''Scaling and Renormalization in Statistical Physics'''<ref name="car96"/>, by John Cardy, 238 pages: an excellent text for understanding the role of CFT in statistical physics, although CFT is not its main subject. * (E) '''Conformal Field Theory'''<ref name="fms97/>, by Philippe di Francesco, Pierre Mathieu and David Sénéchal, 890 pages: the Big Yellow Book on CFT, mostly in 2d, with an in-depth treatment of minimal models and Wess-Zumino-Witten models. == References == {{Reflist|refs= <ref name="prv18">{{cite journal | last=Poland | first=David | last2=Rychkov | first2=Slava | last3=Vichi | first3=Alessandro | title=The conformal bootstrap: Theory, numerical techniques, and applications | journal=Reviews of Modern Physics | publisher=American Physical Society | volume=91 | issue=1 | date=2019-01-11 | issn=0034-6861 | doi=10.1103/revmodphys.91.015002 | doi-access=free | url=https://arxiv.org/abs/1805.04405}}</ref> <ref name="car08">{{cite web | last=Cardy | first=John | title=Conformal Field Theory and Statistical Mechanics | website=arXiv.org | date=2008-07-22 | url=https://arxiv.org/abs/0807.3472 | access-date=2024-11-12}}</ref> <ref name="gin91">{{cite web | last=Ginsparg | first=Paul | title=Applied Conformal Field Theory | website=arXiv.org | date=1988 | url=https://arxiv.org/abs/hep-th/9108028 | access-date=2024-09-03}}</ref> <ref name="ei23">{{cite web | last=Estienne| first=Benoît|last2 = Ikhlef |first2 = Yacine | title=Conformal Field Theory for 2d Statistical Mechanics | date=2023 | url=https://www.lpthe.jussieu.fr/~ikhlef/CFT.pdf | access-date=2024-08-31}}</ref> <ref name="car96">{{cite book | last=Cardy | first=John | title=Scaling and Renormalization in Statistical Physics | publisher=Cambridge University Press | date=1996 | isbn=978-0-521-49959-0 | doi=10.1017/cbo9781316036440 | page=}}</ref> <ref name="rib24">{{cite web | last=Ribault | first=Sylvain | title=Exactly solvable conformal field theories | website=arXiv.org | date=2024-11-26 | url=https://arxiv.org/abs/2411.17262 | access-date=2024-11-27}}</ref> <ref name="fms97>{{cite book | last=di Francesco | first=Philippe | last2=Mathieu | first2=Pierre | last3=Sénéchal | first3=David | title=Conformal Field Theory | publisher=Springer | date=1997 | isbn=0-387-94785-X | page=}}</ref> <ref name="rib14">{{cite web | last=Ribault | first=Sylvain | title=Conformal field theory on the plane | website=arXiv.org | date=2014 | url=https://arxiv.org/abs/1406.4290 | access-date=2024-08-31}}</ref> <ref name="rib16">{{cite journal | last=Ribault | first=Sylvain | title=Minimal lectures on two-dimensional conformal field theory | journal=SciPost Physics Lecture Notes | publisher=Stichting SciPost | date=2018 | url = https://arxiv.org/abs/1609.09523| doi=10.21468/scipostphyslectnotes.1 | doi-access=free | page=}}</ref> <ref name="ryc16">{{cite journal | last= Rychkov|first=Slava |date=2016 |title= EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions| journal=SpringerBriefs in Physics |doi=10.1007/978-3-319-43626-5|arxiv=1601.05000|url = https://arxiv.org/abs/1601.05000 |isbn=978-3-319-43625-8 |s2cid=119192484 }}</ref> }} [[Category:Conformal field theory]] [[Category:CFT course]] fvh2suc8t4zl7zehk9lpilv6vikaiws 0 313557 2690880 2690827 2024-12-08T16:48:52Z Dc.samizdat 2856930 /* Acknowledgements */ 2690880 wikitext text/x-wiki {{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 solely 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-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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 the vertex of a 4-polytope undergoing isoclinic rotation, may also be new in this context. == 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]] 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 explicate a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning understanding of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section entitled ''[[24-cell#Visualization|Visualization]]'' describing the torus decomposition of the 24-cell into cell rings that form discrete Hopf fibrations, previously 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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 solely 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-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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 the vertex of a 4-polytope undergoing isoclinic rotation, may also be new in this context. == 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]] 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 explicate a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning understanding of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section entitled ''[[24-cell#Visualization|Visualization]]'' describing the torus decomposition of the 24-cell into cell rings forming discrete Hopf fibrations, previously 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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-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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 the vertex of a 4-polytope undergoing isoclinic rotation, may also be new in this context. == 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]] 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 explicate a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning understanding of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section entitled ''[[24-cell#Visualization|Visualization]]'' 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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 solely 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-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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning understanding of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section entitled ''[[24-cell#Visualization|Visualization]]'' 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate a hyperspace, such as Euclidean 4-space. Another essential contributor to my dawning understanding of 4-dimensional geometry was Wikipedia editor [[W:User:Cloudswrest|Cloudswrest (A.P. Goucher)]], who authored the section entitled ''[[24-cell#Visualization|Visualization]]'' 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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 any one of the singly curved geodesic circles that are the great circle segments over each {{radic|3}} chord of the rotation.{{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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate 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 entitled ''[[24-cell#Visualization|Visualization]]'' 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate 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 entitled ''[[24-cell#Visualization|Visualization]]'' 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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}} * {{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}} {{Refend}} 093656f3cz6yz5hqhkvijesexze2wgi 2690907 2690903 2024-12-08T21:42:52Z Dc.samizdat 2856930 /* Acknowledgements */ 2690907 wikitext text/x-wiki {{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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate 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]], who authored the section 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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}} * {{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}} {{Refend}} 4t6o7pxseh1pdbldivzbumpf5bfnwrn 2690938 2690930 2024-12-09T01:58:23Z Dc.samizdat 2856930 Undo revision [[Special:Diff/2690930|2690930]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2690938 wikitext text/x-wiki {{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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 explicate 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 rotation, may also be new in this context. == 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]] 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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 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]] 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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]] 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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}} * {{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}} {{Refend}} kww9tjj8vez8oap6mgv9ivauwy4vozi 2690957 2690945 2024-12-09T04:11:10Z Dc.samizdat 2856930 Dc.samizdat moved page [[User:Dc.samizdat/24-cell]] to [[Wikiversity:WikiJournal Preprints/24-cell]]: Submitted for peer review 2690945 wikitext text/x-wiki {{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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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]] 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. 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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 solely 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=; "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."}} 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}} 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: <math display="block">(\pm1, \pm 1, 0, 0) \in \mathbb{R}^4 .</math> 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)<br> 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^o 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: <math display="block">\left( \pm 1, 0, 0, 0 \right)</math> and 16 vertices with ''half-integer'' coordinates of the form: <math display="block">\left( \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2}, \pm \tfrac{1}{2} \right)</math> 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)<br> 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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 12₄16₃ 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 "points" and "lines" 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}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}({{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|5}}(−<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>){{spaces|3}}(−<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>, −<small>{{sfrac|1|2}}</small>,{{spaces|2}}<small>{{sfrac|1|2}}</small>) {{indent|17}}({{spaces|2}}0,{{spaces|2}}0, −1,{{spaces|2}}0)<br> 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° = <small>{{sfrac|{{pi}}|3}}</small> 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° = <small>{{sfrac|{{pi}}|2}}</small> away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = <small>{{sfrac|2{{pi}}|3}}</small> 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° = <small>{{pi}}</small> 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^o 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^o bend, or as a straight line with one 60^o 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 "the convex hull of the neighbouring vertices of a given vertex".{{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 "radius" equals its edge length.{{Efn|The cube is not radially equilateral in Euclidean 3-space <math>\mathbb{R}^3</math>, but a cubic pyramid is radially equilateral in the curved 3-space of the 24-cell's surface, the [[W:3-sphere|3-sphere]] <math>\mathbb{S}^3</math>. 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'' <math>\mathbb{S}^3</math>. In Euclidean 4-space <math>\mathbb{R}^4</math> 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 = 24².{{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=; "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.)".}} 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=: "... 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."}} 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=: "In fact, the vertices of {3,3,5}, each taken 5 times, are the vertices of 25 {3,4,3}'s."}} 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}[120𝛼₄]{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<small>A</small> and Fig 8.2<small>B</small>|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° = <small>{{sfrac|{{pi}}|2}}</small> 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|F₄]], 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=; "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)]".}} 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=; "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 ₀𝑹, ₁𝑹, ₂𝑹, ... determine the polytopes ... whose vertices are the centers of elements 𝐈𝐈₀, 𝐈𝐈₁, 𝐈𝐈₂, ... of the original polytope."}}|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=. "For instance, in the case of <math>\gamma_4[2\beta_4]</math>...."}}{{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=: "Thus the 24 cells of the {3, 4, 3} are dipyramids based on the 24 squares of the <math>\gamma_4</math>. (Their centres are the mid-points of the 24 edges of the <math>\beta_4</math>.)"}} 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=: "...the chess-board has an n-dimensional analogue."}} 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. <!-- 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 --> 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 == 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)]], 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 𝐹₄.}} 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=: "For instance, seeing that the circumference of a circle is 2''π r'', while the surface of a sphere is 4''π r ''², ... 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'' ²''r'' ³."}} 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=; "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."}} 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 "right" and "left", 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 "to the left", the right rotation is not rotating "to the right", 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 "same" directions, or it is left-hand threaded (like a reverse-threaded bolt), moving along the circles in what we conventionally say are "opposite" directions (according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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=; "[[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''."}} 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=: "...the chess-board has an n-dimensional analogue."}} 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 V₀ in the original great hexagon plane of isoclinic rotation P₀, the first vertex reached V₁ is 120 degrees away along a {{radic|3}} chord lying in a different hexagonal plane P₁. P₁ is inclined to P₀ at a 60° angle.{{Efn|P₀ and P₁ 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 V₂ is 120 degrees beyond V₁ along a second {{radic|3}} chord lying in another hexagonal plane P₂ that is Clifford parallel to P₀.{{Efn|P₀ and P₂ 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 V₀ and V₂ are ''two'' {{radic|3}} chords apart,{{Efn|V₀ and V₂ 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}} V₀ and V₂ are ''one'' {{radic|3}} chord apart on some other isocline, and just {{radic|1}} apart on some great hexagon. Between V₀ and V₂, 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).}} P₀ and P₂ are just one {{radic|1}} edge apart (at every pair of ''nearest'' vertices).}} (Notice that V₁ lies in both intersecting planes P₁ and P₂, as V₀ lies in both P₀ and P₁. But P₀ and P₂ have ''no'' vertices in common; they do not intersect.) The third vertex reached V₃ is 120 degrees beyond V₂ along a third {{radic|3}} chord lying in another hexagonal plane P₃ that is Clifford parallel to P₁. V₀ and V₃ 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}} V₀ to V₃ is a 360° isoclinic rotation, and one half of the 24-cell's double-loop hexagram₂ 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 <math>2\pi r</math>; simple rotations of unit-radius polytopes take place on 2𝝅 isoclines. Double rotations may have isoclines of other than <math>2\pi r</math> 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=; "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."}} 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|hexagram₂ 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 hexagram₂ 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 hexagram₂ 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 <math>60^\circ</math> 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|<math>\sqrt 3\approx 1.73</math>.}}}} is a [[W:Skew polygon|skewed]] instance of the [[W:Polygram (geometry)#Regular compound polygons|regular compound polygon]] denoted {6/2}{{=}}2{3} or hexagram₂.{{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=; "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."}} in which each of two "circles" linked in a Möbius "figure eight" 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|octagram₃ 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 "left" 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]] <math>F_a, F_b, F_c, F_d</math>. 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 <math>F_a</math> contains 3 parallel hexagons running clockwise along the cell ring's edges from fibration <math>F_b</math>, and 3 parallel hexagons running counterclockwise along the cell ring's edges from fibration <math>F_c</math>, but that cell ring contains no great hexagons from fibration <math>F_a</math> or fibration <math>F_d</math>. 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]]₂, 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 "halves" 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 "edges" 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 hexagram₂ path. Followed along the column of six octahedra (and "around the end" 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 ''h₁ is {12}, h₂ is {12/5}''}} In contrast, the skew hexagram₂ 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 hexagram₂ 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]] 12₄16₃, 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_8{4}=4{2}]] 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="wikitable" 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]]_3{3/8} ![[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Dodecagons]]_2{12} ![[W:24-gon#Related polygons|24-gram]]_{24/5} ![[#Great squares|Squares]]_6{4} ![[W:24-gon#Related polygons|_{{24/12}={12/2}}]] |- |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 ''h₁'' is {12} }} |In [[600-cell#Hexagons|compounds of 5 24-cells]], isoclines with [[600-cell#Golden chords|golden chords]] of length <big>φ</big> {{=}} {{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 ''h₂'' 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="wikitable floatright" !colspan=6|Characteristics of the 24-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "24-cell"}} |- !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); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>1</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |align=center|<small>120°</small> |align=center|<small><math>\tfrac{2\pi}{3}</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{1}{3}} \approx 0.577</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |- !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|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{12}} \approx 0.289</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</math></small> |- | | | | | |- !align=right|<small><math>_0R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |align=center|<small>45°</small> |align=center|<small><math>\tfrac{\pi}{4}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_1R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{4}} = 0.5</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- !align=right|<small><math>_2R^3/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small> |align=center|<small>30°</small> |align=center|<small><math>\tfrac{\pi}{6}</math></small> |align=center|<small>90°</small> |align=center|<small><math>\tfrac{\pi}{2}</math></small> |- | | | | | |- !align=right|<small><math>_0R^4/l</math></small> |align=center|<small><math>1</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_1R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{3}{4}} \approx 0.866</math></small>{{Efn|name=root 3/4}} |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{2}{3}} \approx 0.816</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_3R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{1}{2}} \approx 0.707</math></small> |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 <small><math>\sqrt{\tfrac{1}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{1}{2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the octahedron), plus <small><math>1</math></small>, <small><math>\sqrt{\tfrac{3}{4}}</math></small>, <small><math>\sqrt{\tfrac{2}{3}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small> (edges which are the characteristic radii of the 24-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{4}}</math></small>, <small><math>\sqrt{\tfrac{1}{12}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{2}}</math></small>, 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|hexagram₂ 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^''q'' R^''r'' 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 Q² is a double rotation (in four dimensions). Every orthogonal transformation is expressible as {{indent|12}}Q^''q'' R^''r''<br> where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as {{indent|12}}Q^''q'' R^''r'' T<br> where 2''q'' + ''r'' + 1 ≤ ''n''.<br> For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q², 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=; "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."}} 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 in the same orientation, 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> and a corresponding set of 16 great hexagon planes represented by quaternion group <math>q8</math>.{{Efn|A quaternion group <math>\pm{q_n}</math> corresponds to a distinct set of Clifford parallel great circle polygons, e.g. <math>q7</math> 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 <math>q7</math>, <math>-q7</math>, <math>q8</math> and <math>-q8</math>.{{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 <math>q_n</math> and <math>-{q_n}</math> generally are distinct sets. The corresponding vertices of the <math>q_n</math> planes and the <math>-{q_n}</math> 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]] <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> to the vertex coordinate <math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math>.{{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 <math>(0,0,1,0)</math>, the Cartesian "north pole". Thus e.g. <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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 <math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> 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="white-space:nowrap;text-align:center" !colspan=15|Proper [[W:SO(4)|rotations]] of the 24-cell [[W:F4 (mathematics)|symmetry group ''F₄'']] {{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 <math>2\pi r</math>, and the former is an isocline of circumference greater than <math>2\pi r</math>.{{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 <math>ql</math>{{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 <math>qr</math> |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,q8}</math><br>[16] 4𝝅 {6/2} |colspan=4|<math>[32]R_{q7,q8}</math>{{Efn|The <math>[32]R_{q7,q8}</math> 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}}<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math>{{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}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: white;"| |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{{Efn|name=four hexagonal fibrations}} 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}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,-q8}</math><br>[16] 4𝝅 {12} |colspan=4|<math>[32]R_{q7,-q8}</math>{{Efn|The <math>[32]R_{q7,-q8}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{-q8}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q7,q7}</math><br>[16] 4𝝅 {1} |colspan=4|<math>[32]R_{q7,q7}</math>{{Efn|The <math>[32]R_{q7,q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q7,-q7}</math><br>[16] 4𝝅 {2} |colspan=4|<math>[32]R_{q7,-q7}</math>{{Efn|The <math>[32]R_{q7,-q7}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=great triangles}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{-q7}</math><br>[16] 2𝝅 {6} |colspan=4|<math>(-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2},-\tfrac{1}{2})</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q7,q1}</math><br>[8] 4𝝅 {12} |colspan=4|<math>[16]R_{q7,q1}</math>{{Efn|The <math>[16]R_{q7,q1}</math> 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|<math>B_4</math> 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)|<math>F_4</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |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 <math>\pm q1</math>, <math>\pm q2</math>, and <math>\pm q3</math>. Each named set{{Efn|Because in the 24-cell each great square is completely orthogonal to another great square, the quaternion groups <math>q1</math> and <math>-{q1}</math> (for example) correspond to the same set of great square planes. That distinct set of 6 disjoint great squares <math>\pm q1</math> 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}}<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/8}=4{6/2}]]{{Efn|name=hexagram}}<br>[[File:Regular_star_figure_4(3,1).svg|100px]]<br><math>^{q7,-q1}</math><br>[8] 4𝝅 {6/2} |colspan=4|<math>[16]R_{q7,-q1}</math>{{Efn|The <math>[16]R_{q7,-q1}</math> 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}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[8] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q1}</math><br>[8] 2𝝅 {4} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: #E6FFEE;"| |{{sfrac|2𝝅|3}} |120° |{{radic|3}} |1.732~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q6,q6}</math><br>[18] 4𝝅 {1} |colspan=4|<math>[36]R_{q6,q6}</math>{{Efn|The <math>[36]R_{q6,q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math>{{Efn|The representative coordinate <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 "reflecting circles") run through cell centers rather than through vertices, and quaternion group <math>q6</math> corresponds to a set of those. However, <math>q6</math> 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 <math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math>. Thus e.g. <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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 <math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q6,-q6}</math><br>[18] 4𝝅 {2} |colspan=4|<math>[36]R_{q6,-q6}</math>{{Efn|The <math>[36]R_{q6,-q6}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q6}</math><br>[18] 2𝝅 {4} |colspan=4|<math>(-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2},0,0)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |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}}<br>[[File:Regular_star_figure_3(4,1).svg|100px]]<br><math>^{q6,-q4}</math><br>[72] 4𝝅 {8/3} |colspan=4|<math>[144]R_{q6,-q4}</math>{{Efn|The <math>[144]R_{q6,-q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q6}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2},0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q4}</math><br>[72] 2𝝅 {4} |colspan=4|<math>(0,0,-\tfrac{\sqrt{2}}{2},-\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |𝝅 |180° |{{radic|4}} |2 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q4,q4}</math><br>[36] 4𝝅 {1} |colspan=4|<math>[72]R_{q4,q4}</math>{{Efn|The <math>[72]R_{q4,q4}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q4}</math><br>[36] 2𝝅 {4} |colspan=4|<math>(0,0,\tfrac{\sqrt{2}}{2},\tfrac{\sqrt{2}}{2})</math> |- style="background: white;"| |2𝝅 |360° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: #E6FFEE;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/2}=2{12}]]{{Efn|name=dodecagon}}<br>[[File:Regular_star_figure_2(6,1).svg|100px]]<br><math>^{q2,q7}</math><br>[48] 4𝝅 {12} |colspan=4|<math>[96]R_{q2,q7}</math>{{Efn|The <math>[96]R_{q2,q7}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[48] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/4}=4{6}]]<br>[[File:Regular_star_figure_4(6,1).svg|100px]]<br><math>^{q7}</math><br>[48] 2𝝅 {6} |colspan=4|<math>(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})</math> |- style="background: #E6FFEE;"| |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|3}} |60° |{{radic|1}} |1 |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,-q2}</math><br>[9] 4𝝅 {2} |colspan=4|<math>[18]R_{q2,-q2}</math>{{Efn|The <math>[18]R_{q2,-q2}</math> 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}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/6}=6{4}]]<br>[[File:Regular_star_figure_6(4,1).svg|100px]]<br><math>^{-q2}</math><br>[9] 2𝝅 {4} |colspan=4|<math>(0,0,0,-1)</math> |- style="background: white;"| |𝝅 |180° |{{radic|4}} |2 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2,q1}</math><br>[12] 4𝝅 {2} |colspan=4|<math>[12]R_{q2,q1}</math>{{Efn|The <math>[12]R_{q2,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q2}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(0,0,0,1)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |{{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="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/1}={24}]]<br>[[File:Regular_polygon_24.svg|100px]]<br><math>^{q1,q1}</math><br>[0] 0𝝅 {1} |colspan=4|<math>[1]R_{q1,q1}</math>{{Efn|The <math>[1]R_{q1,q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[0] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |- style="background: white;"| |0 |0° |{{radic|0}} |0 |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |{{sfrac|𝝅|2}} |90° |{{radic|2}} |1.414~ |- style="background: white;"| |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1,-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>[1]R_{q1,-q1}</math>{{Efn|The <math>[1]R_{q1,-q1}</math> 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}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(1,0,0,0)</math> |rowspan=2|[[W:Icositetragon#Related polygons|{24/12}=12{2}]]<br>[[File:Regular_star_figure_12(2,1).svg|100px]]<br><math>^{-q1}</math><br>[12] 2𝝅 {2} |colspan=4|<math>(-1,0,0,0)</math> |- style="background: white;"| |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 <math>[d]{R_{ql,qr}}</math> each quaternion group <math>\pm{q_n}</math> 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 <math>[4]R_{q7,q8}</math> takes the 4 hexagon planes of <math>q7</math> to the 4 hexagon planes of <math>q8</math> 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 <math>[4]R_{-q7,-q8}</math>, <math>[4]R_{q8,q7}</math> and <math>[4]R_{-q8,-q7}</math>. The name <math>[16]R_{q7,q8}</math> is the conventional representation for all [16] congruent plane displacements. These rotation classes are all subclasses of <math>[32]R_{q7,q8}</math> 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 "same" direction, and a ''left rotation'' is performed by rotating left and right planes in "opposite" directions, according to the [[W:Right hand rule|right hand rule]] by which we conventionally say which way is "up" 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 <math>[32]R_{q7,q8}</math> 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 <math>q7</math> left set and all 4 planes of the <math>q8</math> right set once each.{{Efn|The <math>\pm q7</math> and <math>\pm q8</math> 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 <math>[32]R_{q7,q8}</math> 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]] 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 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.}} 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 collaborative non-profit 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 and recent active research contributions. The original encyclopedia version, the [[Wikipedia:24-cell]] article, is today 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}} == 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}} * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=W:Regular Polytopes (book) }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1991 | title=Regular Complex Polytopes | place=Cambridge | publisher=Cambridge University Press | edition=2nd }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1995 | title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter | publisher=Wiley-Interscience Publication | edition=2nd | isbn=978-0-471-01003-6 | url=https://archive.org/details/kaleidoscopessel0000coxe | editor1-last=Sherk | editor1-first=F. Arthur | editor2-last=McMullen | editor2-first=Peter | editor3-last=Thompson | editor3-first=Anthony C. | editor4-last=Weiss | editor4-first=Asia Ivic | url-access=registration }} ** (Paper 3) H.S.M. Coxeter, ''Two aspects of the regular 24-cell in four dimensions'' ** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] ** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591] ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45] * {{Cite book | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1968 | title=The Beauty of Geometry: Twelve Essays | publisher=Dover | place=New York | edition=2nd }} * {{Cite journal | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1989 | title=Trisecting an Orthoscheme | journal=Computers Math. 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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}} * {{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}} {{Refend}} kww9tjj8vez8oap6mgv9ivauwy4vozi 1 313588 2690959 2679496 2024-12-09T04:11:13Z Dc.samizdat 2856930 Dc.samizdat moved page [[User talk:Dc.samizdat/24-cell]] to [[Wikiversity talk:WikiJournal Preprints/24-cell]]: Submitted for peer review 2679496 wikitext text/x-wiki == Provenance of the content of this article == This version contains only those sections of the [[24-cell|24-cell 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]] 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. It is a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, intended only to gather in one place 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 the document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I appear as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 00:32, 31 October 2024 (UTC) 1pwiijv5q0n4tu80op79rw44cqw1vmi 2690963 2690959 2024-12-09T04:12:20Z Dc.samizdat 2856930 Dc.samizdat moved page [[Wikiversity talk:WikiJournal Preprints/24-cell]] to [[Talk:WikiJournal Preprints/24-cell]]: Misspelled title 2679496 wikitext text/x-wiki == Provenance of the content of this article == This version contains only those sections of the [[24-cell|24-cell 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]] 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. It is a subset of the collaboratively developed [[24-cell|24-cell article]] from which it was extracted, intended only to gather in one place 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 the document may be ascertained precisely by consulting the complete revision history of the [[Wikipedia:24-cell]] article, in which I appear as Wikipedia editor [[Wikipedia:User:Dc.samizdat|Dc.samizdat]]. ([[User talk:Dc.samizdat|discuss]] • [[Special:Contributions/Dc.samizdat|contribs]]) 00:32, 31 October 2024 (UTC) 1pwiijv5q0n4tu80op79rw44cqw1vmi 0 316507 2690858 2690449 2024-12-08T14:05:15Z Sylvain Ribault 2127778 Restructure 2690858 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. However, while it is relatively easy to explain why scale invariance is important in physics, it is not so clear why in many cases scale invariance implies conformal invariance. === The case for scale invariance === === To what extent does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] i7acvzq70s5hr14qvf8hx9m5iprieai 2690859 2690858 2024-12-08T14:14:34Z Sylvain Ribault 2127778 /* Case of flat space */ 2690859 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as changes of scale <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. However, while it is relatively easy to explain why scale invariance is important in physics, it is not so clear why in many cases scale invariance implies conformal invariance. === The case for scale invariance === === To what extent does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] qgu7p72wnxjbnk84y0nrvgxkijglaqj 2690863 2690859 2024-12-08T14:35:30Z Sylvain Ribault 2127778 /* Scale invariance and conformal invariance */ 2690863 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as changes of scale <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. However, while it is relatively easy (but not trivial) to explain why scale invariance is important in physics, it is not so clear why in many cases scale invariance implies conformal invariance. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === The case for scale invariance === === To what extent does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] orebvxmc078kp5oe8hxa6sm9mdm431i 2690895 2690863 2024-12-08T20:23:07Z Sylvain Ribault 2127778 2690895 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as changes of scale <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. However, while it is relatively easy (but not trivial) to explain why scale invariance is important in physics, it is not so clear why in many cases scale invariance implies conformal invariance. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === Scale invariance and phase transitions === === Does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] 148asa1crnj6v3lh6jahiqwjnbfje0v 2690896 2690895 2024-12-08T20:32:33Z Sylvain Ribault 2127778 /* Scale invariance and phase transitions */ 2690896 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as changes of scale <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. However, while it is relatively easy (but not trivial) to explain why scale invariance is important in physics, it is not so clear why in many cases scale invariance implies conformal invariance. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === Scale invariance at phase transitions === === Does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] td8y3xovri2kn3qxjiy64gbunyvfcp9 2690990 2690896 2024-12-09T08:56:20Z Sylvain Ribault 2127778 2690990 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as changes of scale <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. It is relatively easy (but not trivial) to explain why scale invariance is important in physics. Moreover, in many cases scale invariance implies conformal invariance, although the reason is not always clear. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === Scale invariance at phase transitions === === Scale invariance of renormalization group fixed points === Generalize phase transitions. === Does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] cxymbefa135h1nkgw25c8su88ozva59 2690993 2690990 2024-12-09T09:20:15Z Sylvain Ribault 2127778 /* Case of flat space */ 2690993 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as scale transformations <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. It is relatively easy (but not trivial) to explain why scale invariance is important in physics. Moreover, in many cases scale invariance implies conformal invariance, although the reason is not always clear. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === Scale invariance at phase transitions === === Scale invariance of renormalization group fixed points === Generalize phase transitions. === Does scale invariance imply conformal invariance? === == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] 6p9ob1ub55w71fxyklo7r59pg2odm2x 2690994 2690993 2024-12-09T09:30:22Z Sylvain Ribault 2127778 2690994 wikitext text/x-wiki == Conformal transformations == === Definition === On a given space or spacetime <math>M</math> with coordinates <math>x^\mu</math>, distances are defined using a metric <math>g_{\mu\nu}</math>. In particular, the length of an infinitesimal vector <math>v^\mu</math> is <math>\|v\| = \sqrt{g_{\mu\nu}v^\mu v^\nu}</math>. If we know distances, we can also compute angles. The angle <math>\theta</math> between two infinitesimal vectors <math>v^\mu,w^\mu</math> obeys :<math> \cos\theta = \frac{g_{\mu\nu}v^\mu w^\nu}{\sqrt{g_{\mu\nu}v^\mu v^\nu}\sqrt{ g_{\mu\nu}w^\mu w^\nu}} </math> For <math>f:M\to M</math> a diffeomorphism, we define the [[w:pullback]] <math>f^*g</math> of the metric by :<math> (f^*g)_{\mu\nu} = g_{\rho\sigma}(f(x)) \frac{\partial f^\rho}{\partial x^\mu}\frac{\partial f^\sigma}{\partial x^\nu} </math> equivalently <math>(f*g)_{\mu\nu}(x)dx^\mu dx^\nu = g_{\mu\nu}(f)df^\mu df^\nu</math>. A diffeomorphism <math>f:M\to M</math> is called an '''isometry''' if it preserves distances, equivalently if :<math>f^* g = g</math>. It is called a '''conformal transformation''' if it preserves angles. Any isometry is a conformal transformation, but the converse is not true. For any function <math>\lambda:M\to \mathbb{R}</math>, the metrics <math>g_{\mu\nu}</math> and <math>\lambda g_{\mu\nu}</math> define the same angles. So <math>f</math> is a conformal transformation if it preserve the metric up to a scalar factor, :<math> \exists \lambda,\ f^* g = \lambda g </math> The set of conformal transformations is called the '''conformal group''' associated to <math>M,g</math>. Let us indicate how many functions on <math>M</math> are needed to parametrize the objects <math>\lambda, f,g</math>, if <math>d=\dim(M)</math>: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Object ! Number of functions on <math>M</math> |- | Function <math>\lambda:M\to \mathbb{R}</math> | <math>1</math> |- | Diffeomorphism <math>f:M\to M</math> | <math>d</math> |- | Metric <math>g</math> on <math>M</math> | <math>\frac{d(d+1)}{2}</math> |} Therefore, given two metrics <math>g_1,g_2</math>, the condition <math>f^* g_1 = \lambda g_2</math> that they are conformally equivalent involves <math>\frac{d(d+1)}{2}</math> equations for <math>d+1</math> unknowns. For <math>d\leq 2</math>, there is always a solution (under reasonable assumptions): in particular any metric on <math>\mathbb{R}^2</math> is conformally flat. For <math>d\geq 3</math>, two metrics are in general not equivalent modulo rescaling. Similarly, the size of the conformal group depends on the dimension: * For <math>d=1</math>, any diffeomorphism is conformal. * For <math>d\geq 2</math>, the conformal group depends on <math>M,g</math>, with flat space <math>\mathbb{R}^d</math> having the largest possible group <math>SO(d+1,1)</math>. * For <math>d\geq 3</math>, a generic space has only one conformal transformation: the identity. === Case of flat space === The conformal group of <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math> first includes isometries: * Translations. * Rotations. There are also conformal transformations that are not isometries: * Dilations, also known as scale transformations <math>x^\mu \mapsto \lambda x^\mu </math> with <math>\lambda \in \mathbb{R}</math>. * The inversion <math>x^\mu \mapsto \frac{x^\mu}{\|x\|^2}</math>. * Special conformal transformations <math>x^\mu \mapsto \frac{x^\mu - a^\mu \|x\|^2}{1-2a_\mu x^\mu +\|a\|^2\|x\|^2}</math>. These transformations generate the conformal group <math>SO(d+1,1)</math>. === Two-dimensional case === Let <math>M</math> be a two-dimensional connected, oriented manifold. A metric <math>g</math> on <math>M</math> is called Riemannian if it is positive definite. Let <math>\bar g</math> be the equivalence class of <math>g</math> modulo conformal transformations, also called a '''conformal structure'''. Then <math>(M,\bar g)</math> is called a [[w:Riemann surface]]. A Riemann surface is characterized by its topology and its conformal structure. Let us focus on compact Riemann surfaces. Topologically, a compact Riemann surface is characterize by its '''genus''' <math>h\in \mathbb{N}</math>, the number of holes. For a given genus, there is a finite-dimensional '''moduli space''' <math>\mathcal{M}_h</math> of Riemann surfaces. The conformal group of a Riemann surface depends on <math>h</math> and also on the surface: {| class="wikitable" style="font-size:small; text-align:center;" |- ! Genus <math>h</math> ! Name ! <math>\dim_\mathbb{C} \mathcal{M}_h</math> ! Conformal group |- | 0 | Riemann sphere | <math>0</math> | <math>PSL_2(\mathbb{C})</math> |- | 1 | Torus | 1 | <math>\left(\mathbb{C}/\mathbb{Z}^2\right)\rtimes \text{finite} </math> |- | <math>\geq 2</math> | | <math>3h-3</math> | finite |} * The Riemann sphere is conveniently parametrized by a complex coordinate <math>z\in\mathbb{C}\cup \{\infty\}</math>. Its moduli space is trivial, i.e. all genus 0 Riemann surfaces are conformally equivalent. The conformal group is the Möbius group <math>PSL_2(\mathbb{C})</math>, which acts as :<math> z\mapsto \frac{az+b}{cz+d} \qquad \text{for}\qquad \left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{C}) </math> * The torus <math>\mathbb{T} = \frac{\mathbb{C}}{\mathbb{Z}+\tau\mathbb{Z}}</math> is parametrized by a '''modulus''' <math>\tau\in\mathbb{H}\equiv\left\{\tau \in \mathbb{C}|\Im \tau>0\right\}</math>. Two toruses are conformally equivalent if their moduli are related by <math>\tau\mapsto \frac{a\tau+b}{c\tau+d}</math> for <math>\left(\begin{array}{cc} a & b\\ c& d \end{array}\right) \in PSL_2(\mathbb{Z})</math> so the moduli space is <math>\mathcal{M}_1 = \mathbb{H}/PSL_2(\mathbb{Z})</math>. The conformal group of <math>\mathbb{T}</math> includes translations, which are parametrized by <math>\mathbb{T}</math> itself or equivalently by <math>\mathbb{C}/\mathbb{Z}^2</math>. Depending on <math>\tau</math>, there may also be a finite number of rotations. == Conformal symmetry and gravitation == === Conformal invariance and Weyl invariance === In a theory of gravitation such as general relativity, the metric is not fixed, it is a dynamical object. We can therefore combine diffeomorphisms with changes of the metric. In particular, if we combine a diffeomorphism <math>f</math> with <math>g\mapsto f^*g</math>, we obtain a '''change of coordinates''', which should leave the physics invariant. By definition, modulo a change of coordinates, a conformal transformation amounts to a '''Weyl transformation''' of the metric <math>g\mapsto \lambda g</math>. For gravitational theories, conformal invariance is equivalent to Weyl invariance. General relativity is Weyl invariant for <math>d=2</math> only. === Two-dimensional case === In <math>d=2</math>, any field theory of quantum gravity must be a conformal field theory. Moreover, since any metric is conformally flat, we may replace the 3 components of the metric <math>g_{\mu\nu}</math> with the conformal factor <math>\lambda</math>: only 1 bosonic field. Einstein's equation for <math>g_{\mu\nu}</math> then reduces to the Liouville equation for <math>\log \lambda</math>. The Liouville equation is the classical equation of motion of [[w:Liouville theory]]: a solvable CFT that we will study in this course. Therefore, we can build field theories of gravity called '''Liouville gravity''' based on Liouville theory. (There are several possible theories, depending on the matter contents.) === String theory === String theory is a theoretical framework that generalizes quantum field theory, and includes quantum theories of gravity. The basic idea is to describe physics in a spacetime <math>Z</math> using a <math>d=2</math> worldsheet <math>M</math> and an embedding <math>X:M\to Z</math>. The metric on <math>Z</math> induces a metric on <math>M</math>, but it is convenient to allow the worldsheet metric <math>g</math> to be an independent dynamical object, unrelated to the induced metric. Nevertheless, the physics should not depend on <math>g</math>. It is possible to fix 2 of the 3 components of <math>g</math> using a change of coordinates on <math>M</math>, and to get <math>g</math>-independence we also have to take care of the third component, by assuming Weyl invariance. This is why the worldsheet description of string theory uses <math>d=2</math> CFT. String theory provides another motivation for CFT in arbitrary <math>d</math> due to the AdS/CFT correspondence: a holographic relation between a string theory in a <math>d+1</math>-dimensional Anti-de Sitter space, and a CFT in <math>d</math> dimensions. In fact the correspondence can be generalized to non-AdS spaces, which then correspond to non-conformal field theories. Nevertheless, the correspondencs is simplest in the AdS/CFT case. == Scale invariance and conformal invariance == The relevance of conformal field theory extends well beyond gravitational physics. It is relatively easy (but not trivial) to explain why scale invariance is important in physics. Moreover, in many cases scale invariance implies conformal invariance, although the reason is not always clear. Scale invariance is not manifest in physics. Physical properties of matter vary a lot across scales: * At cosmological scales, physics is dominated by gravitational interactions, with important roles for the poorly understood dark matter and dark energy. * In the solar system, Newtonian gravity with relativistic corrections is an adequate description. * From planets to atoms, gravity becomes less and less important, electromagnetism more and more important. These fundamental forces give rise to emergent physical forces such as surface tension. * At nuclear scales, the weak and strong interactions dominate. The strong interaction allows atomic nuclei to hold together in spite of the electromagnetic repulsion between protons. Scale invariance is therefore not a fundamental symmetry of nature. Scale invariance can only hold approximately, over a finite range of scales. Translation invariance is also an approximate symmetry of our universe, but it is an exact symmetry of the fundamental theories. === Scale invariance at phase transitions === Example: Ising Example: water === Scale invariance of renormalization group fixed points === Generalize phase transitions. Renormalize statistical model (Ising). Renormalize field theory. === Does scale invariance imply conformal invariance? === === Universality === More symmetry, fewer parameters (no scale). Critical exponents. Computed by CFT == Exercises == === COGS: The conformal group of flat space === Consider the Euclidean space <math>\mathbb{R}^d</math> with the flat metric <math>g_{\mu\nu}=\delta_{\mu\nu}</math>, and the Minkowski space <math>\mathbb{R}^{d+1,1}</math> with coordinates <math>Y=\left(y^\mu,y^-,y^+\right)</math> with <math>\mu = 1,2,\dots, d</math> and the flat metric <math>\|dY\|^2 = \sum_{\mu=1}^d \left(dy^\mu\right)^2 -dy^-dy^+ </math>. Consider the diffeormorphisms :<math> \varphi:\left\{\begin{array}{ccl} \mathbb{R}^d & \to & \mathbb{R}^{d+1,1} \\ x^\mu &\mapsto & \left(x^\mu,\|x\|^2,1\right) \end{array}\right. \quad , \quad \psi: \left\{\begin{array}{ccl} \mathbb{R}^{d+1,1} & \to & \mathbb{R}^{d+1,1} \\ Y &\mapsto & \frac{1}{y^+}Y = \left(\frac{y^\mu}{y^+},\frac{y^-}{y^+},1\right) \end{array}\right. </math> # Check that <math>\varphi</math> is an isometry. Is <math>\psi</math> an isometry? Is it a conformal transformation? # Show that the restriction of <math>\psi</math> to the light cone <math>\mathcal{L}=\left\{Y\in \mathbb{R}^{d+1,1}\left| \|Y\|^2 = 0\right.\right\}</math> is a conformal transformation, and that <math>\varphi(\mathbb{R}^d)\subset \mathcal{L}</math>. # Let <math>G\in SO(d+1,1)</math> be an isometry of <math>\mathbb{R}^{d+1,1}</math>, in particular <math>G</math> is linear. Show that <math>\varphi^{-1}\circ \psi \circ G\circ \varphi</math> is a conformal transformation of <math>\mathbb{R}^d</math>. Deduce that the conformal group of <math>\mathbb{R}^d</math> includes <math>SO(d+1,1)</math>. # Explicitly write the action of <math>G\in SO(d+1,1)</math> on <math>x^\mu</math>. [[Category: CFT course]] rk8d0iizjmdsmh2s66aget90n8sa0h9 2 316552 2690932 2690655 2024-12-09T01:39:45Z Jaredscribe 2906761 /* Personnel */ Transition spokesman Brian Hughes said .. list of 10+ 2690932 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies. Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} pyoyt848eh001ok2gbe5v9dfgzlm1rq 2690933 2690932 2024-12-09T01:41:08Z Jaredscribe 2906761 /* Personnel */ NYT 2690933 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} e77drtun02idpi9fjqculxbnv4azxzx 2690934 2690933 2024-12-09T01:43:33Z Jaredscribe 2906761 /* Personnel */ 2690934 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1<nowiki> Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 5l7w429wg3hl6d4x2vphvbgywlxzobi 2690935 2690934 2024-12-09T01:46:02Z Jaredscribe 2906761 reorganize 2690935 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 0gux68xch9s5lbh9ks2yre48v2u9spu 2690936 2690935 2024-12-09T01:54:42Z Jaredscribe 2906761 /* U.S. Department of Health and Human Services */ National Institutes of Health Nominated to lead the National Institutes of Health is Jay Bhattacharya. 2690936 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} rkf8a3q5spe6gsiy5lsww7d0iy7g7vl 2690937 2690936 2024-12-09T01:56:35Z Jaredscribe 2906761 /* National Institutes of Health */ 2690937 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. The NIH distributes grants of ~$50bn per year. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} ig4rwapjgvm8wvqlhcbmsy0t3pg1ffw 2690942 2690937 2024-12-09T02:17:15Z Jaredscribe 2906761 /* National Institutes of Health */ 2690942 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. The NIH distributes grants of ~$50bn per year. Mr. Battacharya has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic". * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]] Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} r7caijddifbr6thtkst14zxstto6q7y 2690943 2690942 2024-12-09T02:20:27Z Jaredscribe 2906761 /* National Institutes of Health */ refocusing on chronic diseases, whose research is underfunded, away from infectious diseases, whose research is overfunded. ending gain-of-function research. 2690943 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo=DOGE Logo as of November 14, 2024.jpg |logo_size=150px |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} }} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. The NIH distributes grants of ~$50bn per year. Mr. Battacharya has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic". * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]] * refocusing on [[w:Chronic_diseases|chronic diseases]], whose research is underfunded, away from [[w:Infectious_diseases|infectious diseases]], whose research is overfunded. * ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} p4q9o8ukakdnfs50wz6kent9xo9mpiq 2690946 2690943 2024-12-09T02:37:05Z Jaredscribe 2906761 Small-government advocate Ron Paul has suggested to cut aid to the following "biggest" welfare recipients: The Military-industrial complex The Pharmaceutical-industrial complex The Federal Reserve To which Mr. Musk replied, "Needs to be done".[1] 2690946 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. The NIH distributes grants of ~$50bn per year. Mr. Battacharya has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic". * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]] * refocusing on [[w:Chronic_diseases|chronic diseases]], whose research is underfunded, away from [[w:Infectious_diseases|infectious diseases]], whose research is overfunded. * ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} 2p8bmv7s842dk8a2ufc8r3tfzw8iufe 2690947 2690946 2024-12-09T02:59:48Z Jaredscribe 2906761 2690947 wikitext text/x-wiki {{Research project}} The U.S. [[w:Department of Government Efficiency]]. {{Infobox Organization |name=Department of Government Efficiency |logo= |logo_size= |logo_caption=Logo on [[Twitter|X]] (formerly Twitter) as of November 14, 2024 |seal= |seal_size= |seal_caption= |formation=Announced on November 12, 2024; yet to be established |abbreviation=DOGE |key_people={{plainlist|[[w:Commissioner of the Department of Government Efficiency|Co-commissioners]]: * [[w:Elon Musk]] * [[w:Vivek Ramaswamy]] }} |website={{URL|https://x.com/DOGE|x.com/DOGE}} |volunteers=* Federico Sturzenegger|services=consulting|headquarters=Mar-A-Lago|organization_type=Presidential Advisory Commission|founder=Donald Trump|extinction=4 July 2026 (planned)|mission=(In the words of president-elect Donald Trump: * dismantle government bureaucracy * slash excess regulations * cut wasteful expenditures * restructure federal agencies, * address "massive waste and fraud" in government spending}} This "'''Wiki Of Government Efficiency'''" (WOGE) is a public interest, non-partisan research project that will [[User:Jaredscribe/Department of Government Efficiency#Reduce the deficit and debt by impounding appropriated funds|analyze the U.S. federal budget]], [[User:Jaredscribe/Department of Government Efficiency#Reform the other Government Bureaus and Departments|federal bureaucracy]], and [[User:Jaredscribe/Department of Government Efficiency#Shrink the federal civil service|federal civil service]], in the context of [[w:Second_presidency_of_Donald_Trump|president-elect Trump']]<nowiki/>s [[w:Agenda_47|Agenda 47]], and will catalogue, evaluate, and critique proposals on how the '''[[w:Department of Government Efficiency|Department of Government Efficiency]]'''{{Efn|Also referred to as '''Government Efficiency Commission'''}} (DOGE) is or is not fulfilling its mission to ''"dismantle government bureaucracy, slash excess regulations, and cut wasteful expenditures and restructure federal agencies"'', in the words of president-elect [[wikipedia:Donald_Trump|Donald Trump]], who called for it to address ''"massive waste and fraud"'' in government spending.<ref name=":1">{{Cite web|url=https://www.bbc.co.uk/news/articles/c93qwn8p0l0o|title=Donald Trump picks Elon Musk for US government cost-cutting role|last1=Faguy|first1=Ana|last2=FitzGerald|first2=James|date=2024-11-13|publisher=BBC News|language=en-GB|access-date=2024-11-13}}</ref> Here's [[User:Jaredscribe/Department of Government Efficiency/How to contribute|how to contribute]] to the WOGE. The DOGE intends to [[User:Jaredscribe/Department of Government Efficiency#Office of Management and Budget|work through the Office of Management and Budget]] as its "policy vector". The [[w:U.S._budget_deficit|U.S. Budget deficit]], (C.f. [[w:Government_budget_balance|fiscal deficit]]), and the [[w:National_debt_of_the_United_States|U.S. National debt]], currently $35.7 Trillion as of 10/2024, which is 99% of the [[w:U.S._GDP|U.S. GDP]],<ref>{{Unbulleted list citebundle|{{cite news|newspaper=Financial Post| title= Musk's $2 Trillion of Budget Cuts Would Have These Stocks Moving|url=https://financialpost.com/pmn/business-pmn/musks-2-trillion-of-budget-cuts-would-have-these-stocks-moving|first=Alexandra|last=Semenova|date=November 4, 2024}}|{{cite news|newspaper= New York Times|title=Elon|url=https://nytimes.com/2024/10/29/us/politics/elon-musk-trump-economy-hardship.html}}|{{Cite web |date=September 5, 2024 |title=Trump says he'd create a government efficiency commission led by Elon Musk |url=https://apnews.com/article/donald-trump-elon-musk-government-efficiency-commission-e831ed5dc2f6a56999e1a70bb0a4eaeb |publisher=AP News}}|{{cite web|first=Jenn|last=Brice|title=How Elon Musk's $130 million investment in Trump's victory could reap a huge payoff for Tesla and the rest of his business empire|url=https://fortune.com/2024/11/06/elon-musk-donald-trump-tesla-spacex-xai-boring-neuralink|website=Fortune}}|{{cite web|url=https://axios.com/2024/11/07/elon-musk-government-efficiency-trump|title=Musk will bring his Twitter management style to government reform}}|{{cite news| access-date =November 9, 2024|work=Reuters|date=September 6, 2024|first1=Helen|first2=Gram|last1=Coster| last2=Slattery|title=Trump says he will tap Musk to lead government efficiency commission if elected| url= https://reuters.com/world/us/trump-adopt-musks-proposal-government-efficiency-commission-wsj-reports-2024-09-05}}|{{cite web|title=Trump says Musk could head 'government efficiency' force|url= https://bbc.com/news/articles/c74lgwkrmrpo|publisher=BBC}}|{{cite web|date =November 5, 2024|title=How Elon Musk could gut the government under Trump|url=https://independent.co.uk/news/world/americas/us-politics/elon-musk-donald-trump-economy-job-cuts-b2641644.html|website= The Independent}}}}</ref> and expected to grow to 134% of GDP by 2034 if current laws remain unchanged, according to the [[w:Congressional_Budget_Office|Congressional Budget Office]]. The DOGE will be a [[wikipedia:Presidential_commission_(United_States)|presidential advisory commission]] led by the billionaire businessmen [[wikipedia:Elon_Musk|Elon Musk]] and [[wikipedia:Vivek_Ramaswamy|Vivek Ramaswamy]], and possibly [[w:Ron_Paul|Ron Paul]]<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=November 5, 2024|website=The Hill}}</ref><ref>{{Cite web|url=https://usatoday.com/story/business/2024/10/28/patricia-healy-elon-musk-highlights-need-for-government-efficiency/75798556007|title=Elon Musk puts spotlight on ... Department of Government Efficiency? {{!}} Cumberland Comment|last=Healy|first=Patricia|website=USA TODAY|language=en-US|access-date=November 9, 2024}}</ref>. Musk stated his belief that DOGE could remove US$2 trillion from the [[w:United_States_federal_budget|U.S. federal budget]],<ref>{{Cite web|url=https://www.youtube.com/live/HysDMs2a-iM?si=92I5LD1FY2PAsSuG&t=15822|title=WATCH LIVE: Trump holds campaign rally at Madison Square Garden in New York|date=October 28, 2024|website=youtube.com|publisher=[[PBS NewsHour]]|language=en|format=video}}</ref> without specifying whether these savings would be made over a single year or a longer period.<ref>{{Cite web|url=https://www.bbc.co.uk/news/articles/cdj38mekdkgo|title=Can Elon Musk cut $2 trillion from US government spending?|last=Chu|first=Ben|date=2024-11-13|website=BBC News|language=en-GB|access-date=2024-11-14}}</ref> [[File:2023_US_Federal_Budget_Infographic.png|thumb|An infographic on outlays and revenues in the 2023 [[United States federal budget|U.S. federal budget]]]] DOGE could also streamline permitting with “categorical exclusions” from environmental reviews under the National Environmental Policy Act. {{sidebar with collapsible lists|name=U.S. deficit and debt topics|namestyle=background:#bf0a30;|style=width:22.0em; border: 4px double #d69d36; background:var(--background-color-base, #FFFFFF);|bodyclass=vcard|pretitle='''<span class="skin-invert">This article is part of [[:Category:United States|a series]] on the</span>'''|title=[[United States federal budget|<span style="color:var(--color-base, #000000);">Budget and debt in the<br/>United States of America</span>]]|image=[[File:Seal of the United States Congress.svg|90px]] [[File:Seal of the United States Department of the Treasury.svg|90px]]|titlestyle=background:var(--background-color-base, #002868); background-clip:padding-box;|headingstyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff);|listtitlestyle=background:var(--background-color-base, #bf0a30); color:var(--color-base, #fff); text-align:center;|expanded={{{expanded|{{{1|}}}}}}|list1name=dimensions|list1title=Major dimensions|list1class=hlist skin-invert|list1=* [[Economy of the United States|Economy]] * [[Expenditures in the United States federal budget|Expenditures]] * [[United States federal budget|Federal budget]] * [[Financial position of the United States|Financial position]] * [[Military budget of the United States|Military budget]] * [[National debt of the United States|Public debt]] * [[Taxation in the United States|Taxation]] * [[Unemployment in the United States|Unemployment]] * [[Government_spending_in_the_United_States|Gov't spending]]|list2name=programs|list2title=Programs|list2class=hlist skin-invert|list2=* [[Medicare (United States)|Medicare]] * [[Social programs in the United States|Social programs]] * [[Social Security (United States)|Social Security]]|list3name=issues|list3title=Contemporary issues|list3class=skin-invert|list3=<div style="margin-bottom:0.5em"> [[National Commission on Fiscal Responsibility and Reform|Bowles–Simpson Commission]] {{flatlist}} * <!--Bu--> [[Bush tax cuts]] * <!--Deb--> [[United States debt ceiling|Debt ceiling]] ** [[History of the United States debt ceiling|history]] * <!--Def--> [[Deficit reduction in the United States|Deficit reduction]] * <!--F--> [[United States fiscal cliff|Fiscal cliff]] * <!--H--> [[Healthcare reform in the United States|Healthcare reform]] * <!--P--> [[Political debates about the United States federal budget|Political debates]] * <!--So--> [[Social Security debate in the United States|Social Security debate]] * <!--St--> "[[Starve the beast]]" * <!--Su--> [[Subprime mortgage crisis]] {{endflatlist}} </div> [[2007–2008 financial crisis]] {{flatlist}} * <!--D--> [[United States debt-ceiling crisis (disambiguation)|Debt-ceiling crises]] ** [[2011 United States debt-ceiling crisis|2011]] ** [[2013 United States debt-ceiling crisis|2013]] ** [[2023 United States debt-ceiling crisis|2023]] {{endflatlist}} [[2013 United States budget sequestration|2013 budget sequestration]] {{flatlist}} * <!--G--> [[Government shutdowns in the United States|Government shutdowns]] ** [[1980 United States federal government shutdown|1980]] ** [[1981, 1984, and 1986 U.S. federal government shutdowns|1981, 1984, 1986]] ** [[1990 United States federal government shutdown|1990]] ** [[1995–1996 United States federal government shutdowns|1995–1996]] ** [[2013 United States federal government shutdown|2013]] ** [[January 2018 United States federal government shutdown|Jan 2018]] ** [[2018–2019 United States federal government shutdown|2018–2019]] {{endflatlist}} Related events {{flatlist}} *<!--E-->[[Removal of Kevin McCarthy as Speaker of the House|2023 Removal of Kevin McCarthy]] {{endflatlist}}|list4name=terminology|list4title=Terminology|list4class=hlist skin-invert|list4=Cumulative [[Government budget balance|deficit]] + [[National debt of the United States#Debates|Interest]] ≈ [[Government debt|Debt]] * [[Balance of payments]] * [[Inflation]] * [[Continuing resolution]]}} [[w:Deficit_reduction_in_the_United_States|Deficit reduction in the United States]] == Deregulate the Economy == The legal theory that this can be done through the executive branch is found in the U.S. Supreme Court’s ''[[w:West_Virginia_v._EPA|West Virginia v. EPA]]'' and ''[[w:Loper_Bright|Loper Bright]]'' rulings, which rein in the administrative state and mean that much of what the federal government now does is illegal.<ref>{{cite web|url=https://www.wsj.com/opinion/department-of-government-efficiency-elon-musk-vivek-ramaswamy-donald-trump-1e086dab|website=[[w:Wall Street Journal]]|title=The Musk-Ramaswamy Project Could Be Trump’s Best Idea}}</ref> Mr. Trump has set a goal of eliminating 10 regulations for every new one. The [[w:Competitive_Enterprise_Institute|Competitive Enterprise Institute]]’s Wayne Crews says 217,565 rules have been issued since the [[w:Federal_Register|Federal Register]] first began itemizing them in 1976, with 89,368 pages added last year. [https://sgp.fas.org/crs/misc/R43056.pdf 3,000-4,500 rules are added each year]. DOGE’s first order will be to pause enforcement of overreaching rules while starting the process to roll them back. Mr. Trump and DOGE could direct agencies to settle legal challenges to Biden rules by vacating them. This could ease the laborious process of undoing them by rule-making through the [[w:Administrative_Procedure_Act|Administrative Procedure Act]]. A source tells the WSJ they’ll do whatever they think they legally can without the APA. The [[w:Congressional_Review_Act|Congressional Review Act]]—which allows Congress to overturn recently issued agency regulations—had been used only once, prior to [[w:First_presidency_of_Donald_Trump|Trump's first term]]. While in office, he and the Republican Congress used it on 16 rules. This time, there will be more than 56 regulatory actions recent enough to be repealed. The [[w:Chevron_deference|''Chevron'' deference]] had required federal courts to defer to agencies’ interpretations of ambiguous statutes, but this was overturned in 2024. Taken together, with some other recent [[w:SCOTUS|SCOTUS]] rulings, we now have, says the WSJ, the biggest opportunity to cut regulatory red tape in more than 40 years.<ref>[https://www.wsj.com/opinion/let-the-trump-deregulation-begin-us-chamber-of-commerce-second-term-economic-growth-73f24387?cx_testId=3&cx_testVariant=cx_166&cx_artPos=0]</ref><blockquote>"Most legal edicts aren’t laws enacted by Congress but “rules and regulations” promulgated by unelected bureaucrats—tens of thousands of them each year. Most government enforcement decisions and discretionary expenditures aren’t made by the democratically elected president or even his political appointees but by millions of unelected, unappointed civil servants within government agencies who view themselves as immune from firing thanks to civil-service protections." "This is antidemocratic and antithetical to the Founders’ vision. It imposes massive direct and indirect costs on taxpayers." "When the president nullifies thousands of such regulations, critics will allege executive overreach. In fact, it will be ''correcting'' the executive overreach of thousands of regulations promulgated by administrative fiat that were never authorized by Congress. The president owes lawmaking deference to Congress, not to bureaucrats deep within federal agencies. The use of executive orders to substitute for lawmaking by adding burdensome new rules is a constitutional affront, but the use of executive orders to roll back regulations that wrongly bypassed Congress is legitimate and necessary to comply with the Supreme Court’s recent mandates. And after those regulations are fully rescinded, a future president couldn’t simply flip the switch and revive them but would instead have to ask Congress to do so"<ref>{{Cite news|url=https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5|title=Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government|last=Musk|first=Elon|date=20 November 2024|work=The Wall Street Journal|last2=Ramaswamy|first2=Vivek}}</ref> </blockquote> == Shrink the federal civil service == The government has around three million [[w:United_States_federal_civil_service|federal civil service]] employees, with an average salary of $106,000. Dr. Anthony Fauci made $481,000 in 2022. The federal head count has ballooned by 120,800 during the Biden years. Civil service and union protections make it hard to fire workers. Mr. Trump intends to quickly resurrect the [[w:Schedule_F|Schedule F]] reform that he sought to implement at the end of his first term but was scrapped by Mr. Biden. These would high-level federal employees to be removed like political appointees, by eliminating their job protections. WSJ proposals[https://www.wsj.com/opinion/the-doge-cheat-sheet-elon-musk-vivek-ramaswamy-department-of-government-efficiency-1c231783#cxrecs_s] The [[w:Administrative_Procedures_Act|Administrative Procedures Act]] statute protects federal employees from political retaliation, but allows for “reductions in force” that don’t target specific employees. The statute further empowers the president to “prescribe rules governing the competitive service.” The Supreme Court has held—in ''[[w:Franklin_v._Massachusetts|Franklin v. Massachusetts]]'' (1992) and ''[[w:Collins_v._Yellen|Collins v. Yellen]]'' (2021) that when revious presidents have used this power to amend the civil service rules by executive order, they weren’t constrained by the APA when they did so. Mr. Trump can, with this authority, implement any number of “rules governing the competitive service” that would curtail administrative overgrowth, from large-scale firings to relocation of federal agencies out of the Washington area. The DOGE welcomes voluntary terminations once the President begins requiring federal employees to come to the office five days a week, because American taxpayers shouldn’t pay federal employees for the Covid-era privilege of staying home.<ref>[https://www.wsj.com/opinion/musk-and-ramaswamy-the-doge-plan-to-reform-government-supreme-court-guidance-end-executive-power-grab-fa51c020?cx_testId=3&cx_testVariant=cx_165&cx_artPos=5 Elon Musk and Vivek Ramaswamy: The DOGE Plan to Reform Government: Following the Supreme Court’s guidance, we’ll reverse a decadeslong executive power grab. Musk & Ramaswamy 11/20/2024]</ref> == Reduce the deficit and debt by impounding appropriated funds == === Impound appropriated funds === Reports suggest that president-elect Trump intends to override Congress’s power of the purse by [[w:Impoundment_of_appropriated_funds|impoundment of appropriated funds]], that is, refusing to spend them. the president may [[wikipedia:Rescission_bill|propose rescission]] of specific funds, but that rescission must be approved by both the [[wikipedia:United_States_House_of_Representatives|House of Representatives]] and [[wikipedia:United_States_Senate|Senate]] within 45 days. [[w:Thomas_Jefferson|Thomas Jefferson]] was the first president to exercise the power of impoundment in 1801, which power was available to all presidents up to and including [[wikipedia:Richard_Nixon|Richard Nixon]], and was regarded as a power inherent to the office, although one with limits. He may ask Congress to repeal The [[w:Congressional_Budget_and_Impoundment_Control_Act_of_1974|Congressional Budget and Impoundment Control Act of 1974]], which was passed in response to Nixon's abuses.<ref>{{Cite web|url=http://democrats-budget.house.gov/resources/reports/impoundment-control-act-1974-what-it-why-does-it-matter|title=The Impoundment Control Act of 1974: What Is It? Why Does It Matter? {{!}} House Budget Committee Democrats|date=2019-10-23|website=democrats-budget.house.gov|language=en|access-date=2024-05-19}}</ref> If Congress refuses to do so, president Trump may impound funds anyway and argue in court that the 1974 law is unconstitutional. The matter would likely end up at the Supreme Court, which would have to do more than simply hold the 1974 act unconstitutional in order for Mr. Trump to prevail. The court would also have to overrule [[w:Train_v._City_of_New_York_(1975)|''Train v. City of New York'' (1975)]], which held that impoundment is illegal unless the underlying legislation specifically authorizes it. === Reduce the budget deficit === [[wikipedia:U.S. federal budget|U.S. federal budget]] The [[wikipedia:Fiscal_year|fiscal year]], beginning October 1 and ending on September 30 of the year following. Congress is the body required by law to pass appropriations annually and to submit funding bills passed by both houses to the President for signature. Congressional decisions are governed by rules and legislation regarding the [[wikipedia:United_States_budget_process|federal budget process]]. Budget committees set spending limits for the House and Senate committees and for Appropriations subcommittees, which then approve individual [[wikipedia:Appropriations_bill_(United_States)|appropriations bills]] to During FY2022, the federal government spent $6.3 trillion. Spending as % of GDP is 25.1%, almost 2 percentage points greater than the average over the past 50 years. Major categories of FY 2022 spending included: Medicare and Medicaid ($1.339T or 5.4% of GDP), Social Security ($1.2T or 4.8% of GDP), non-defense discretionary spending used to run federal Departments and Agencies ($910B or 3.6% of GDP), Defense Department ($751B or 3.0% of GDP), and net interest ($475B or 1.9% of GDP).<ref name="CBO_2022">[https://www.cbo.gov/publication/58888 The Federal Budget in Fiscal Year 2022: An Infographic]</ref> CBO projects a federal budget deficit of $1.6 trillion for 2024. In the agency’s projections, deficits generally increase over the coming years; the shortfall in 2034 is $2.6 trillion. The deficit amounts to 5.6 percent of gross domestic product (GDP) in 2024, swells to 6.1 percent of GDP in 2025, and then declines in the two years that follow. After 2027, deficits increase again, reaching 6.1 percent of GDP in 2034.<ref name="CBO_budgetOutlook2024">{{cite web|url=https://www.cbo.gov/publication/59710|title=The Budget and Economic Outlook: 2024 to 2034|date=February 7, 2024|publisher=CBO|access-date=February 7, 2024}}</ref> The following table summarizes several budgetary statistics for the fiscal year 2015-2021 periods as a percent of GDP, including federal tax revenue, outlays or spending, deficits (revenue – outlays), and [[wikipedia:National_debt_of_the_United_States|debt held by the public]]. The historical average for 1969-2018 is also shown. With U.S. GDP of about $21 trillion in 2019, 1% of GDP is about $210 billion.<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> Statistics for 2020-2022 are from the CBO Monthly Budget Review for FY 2022.<ref name="CBO_MBRFY2022">{{cite web|url=https://www.cbo.gov/publication/58592|title=Monthly Budget Review: Summary for Fiscal Year 2022|date=November 8, 2022|publisher=CBO|access-date=December 10, 2022}}</ref> {| class="wikitable" !Variable As % GDP !2015 !2016 !2017 !2018 !2019 !2020 !2021 !2022 !Hist Avg |- !Revenue<ref name="CBO_Hist_20">[https://www.cbo.gov/about/products/budget-economic-data#2 CBO-Historical Budget Data-Retrieved January 28, 2020]</ref> |18.0% |17.6% |17.2% |16.4% |16.4% |16.2% |17.9% |19.6% |17.4% |- !Outlays<ref name="CBO_Hist_20" /> |20.4% |20.8% |20.6% |20.2% |21.0% |31.1% |30.1% |25.1% |21.0% |- !Budget Deficit<ref name="CBO_Hist_20" /> | -2.4% | -3.2% | -3.5% | -3.8% | -4.6% | -14.9% | -12.3% | -5.5% | -3.6% |- !Debt Held by Public<ref name="CBO_Hist_20" /> |72.5% |76.4% |76.2% |77.6% |79.4% |100.3% |99.6% |94.7% | |} The [[wikipedia:U.S._Constitution|U.S. Constitution]] ([[wikipedia:Article_One_of_the_United_States_Constitution|Article I]], section 9, clause 7) states that "No money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time." Each year, the President of the United States submits a budget request to Congress for the following fiscal year as required by the [[wikipedia:Budget_and_Accounting_Act_of_1921|Budget and Accounting Act of 1921]]. Current law ({{UnitedStatesCode|31|1105}}(a)) requires the president to submit a budget no earlier than the first Monday in January, and no later than the first Monday in February. Typically, presidents submit budgets on the first Monday in February. The budget submission has been delayed, however, in some new presidents' first year when the previous president belonged to a different party. === Reduce the National debt === == Strategic Foreign Policy and Military reform == === Department of State === {{Main article|w:Second_presidency_of_Donald_Trump#Prospective_foreign_policy|w:State Department}} === U.S. Department of Defense === The president-elect is reportedly considering a draft executive order that establishes a “warrior board” of retired senior military personnel with the power to review three- and four-star officers “on leadership capability, strategic readiness, and commitment to military excellence," and to recommend removals of any deemed unfit for leadership. This would fast-track the removal of generals and admirals found to be “lacking in requisite leadership qualities,” consistent with his earlier vow to fire “woke” military leaders.<ref>[https://www.wsj.com/politics/national-security/trump-draft-executive-order-would-create-board-to-purge-generals-7ebaa606<nowiki> Trump draft executive order would create a board to purge generals 11/12/2024]</nowiki></ref> There are legal obstacles. The law prohibits the firing of commissioned officers except by “sentence of a general court-martial,” as a “commutation of a sentence of a general court-martial,” or “in time of war, by order of the president.” A commissioned officer who believes he’s been wrongfully dismissed has the right to seek a trial by court-martial, which may find the dismissal baseless. <ref>[https://www.wsj.com/opinion/trump-tests-the-constitutions-limits-checks-balances-government-policy-law-78d0d0f1 &lt;nowiki&gt; Trump test the constitutions limits 11/19/2024]</ref> Musk said, "Some idiots are still building manned fighter jets like the F-35," and later added: "Manned fighter jets are outdated in the age of drones and only put pilots' lives at risk." [[w:Bernie_Sanders|Bernie Sanders]] wrote on X: "Elon Musk is right. The Pentagon, with a budget of $886 billion, just failed its 7th audit in a row. It's lost track of billions. Last year, only 13 senators voted against the Military Industrial Complex and a defense budget full of waste and fraud. That must change."{{sfn|Newsweek 12/02|2024}}. It failed its fifth audit in June 2023.<ref>{{Cite web|url=https://www.newsweek.com/fox-news-host-confronts-gop-senator-pentagons-fifth-failed-audit-1804379|title=Fox News host confronts GOP Senator on Pentagon's fifth failed audit|last=Writer|first=Fatma Khaled Staff|date=2023-06-04|website=Newsweek|language=en|access-date=2024-12-02}}</ref> === US Space Force === The [[w:US_Space_Force|US Space Force]]'s 2023 budget was ~$26bn and it had 9,400 military personnel. SpaceX had a $14m contract to provide communications to the Ukrainian armed forces and government through 30th Nov 2024.{{sfn|Economist 11/23|2024}} Is also receiving a $733m contract to carry satellites into orbit.{{sfn|Economist 11/23|2024}} The Pentagon plans to incorporate into its own communications network 100 of [[w:Starshield|Starshield]]'s satellites.{{sfn|Economist 11/23|2024}} Starshield also has a $1.8bn contract to help the [[w:National_Reconnaissance_Office|National Reconnaissance Office]] build spy satellites.{{sfn|Economist 11/23|2024}} == NASA == The [[w:National_Air_and_Space_Administration|National Air and Space Administration]] had a 2023 budget of $25.4 bn and 18,000 employees. [[w:Jared_Isaacman|Jared Isaacman]] is nominated director. He had joined a space voyage in 2021 which was the first for an all-civilian crew to reach orbit. He led a four person crew in September on the first commericial spacewalk, testing SpaceX's new spacesuits. He promised to lead NASA in to "usher in an era where humanity becomes a true space-faring civilization."<ref>{{Cite news|url=https://www.wsj.com/politics/elections/trump-picks-billionaire-space-traveler-to-run-nasa-4420150b?page=1|title=Trump Picks Billionaire Space Traveler to Run NASA|last=Maidenberg|first=Micah|date=5 December 2024|work=WSJ}}</ref> In September 2026, NASA's [[w:Artemis_program|Artemis program]], established in 2017 via [[wikipedia:Space_Policy_Directive_1|Space Policy Directive 1]], is intended to reestablish a human presence on the Moon for the first time since the [[wikipedia:Apollo_17|Apollo 17]] mission in 1972. The program's stated long-term goal is to establish a [[wikipedia:Moonbase|permanent base on the Moon]] to facilitate [[wikipedia:Human_mission_to_Mars|human missions to Mars]]. The [[w:U.S._National_Academies_of_Sciences,_Engineering,_and_Medicine|U.S. National Academies of Sciences, Engineering, and Medicine]] in October, put out a report titled "NASA at a Crossroads," which identified myriad issues at the agency, including out-of-date infrastructure, pressures to prioritize short-term objectives and inefficient management practices. NASA's costly [[w:Space_Launch_System|Space Launch System]] (SLS) is the cornerstone of the Artemis program. has a price tag of around $4.1 billion per launch, and is a single-use rocket that can only launch every two years, having debuted in 2022 with the uncrewed [[w:Artemis_1_mission|Artemis 1 mission]] to the moon. In contrast, SpaceX is working to reduce the cost of a single Starship flight to under $10 million. NASA Associate Administrator Jim Free urged the incoming administration to maintain the current plans, in a symposium with the [[w:American_Astronautical_Society|American Astronautical Society]] saying "We need that consistency in purpose. That has not happened since Apollo. If we lose that, I believe we will fall apart and we will wander, and other people in this world will pass us by." NASA has already asked both [[w:SpaceX|SpaceX]] and also Jeff Bezos' [[w:Blue_Origin|Blue Origin]], to develop cargo landers for its Artemis missions and to deliver heavy equipment on them to the Moon by 2033. "Having two lunar lander providers with different approaches for crew and cargo landing capability provides mission flexibility while ensuring a regular cadence of moon landings for continued discovery and scientific opportunity," Stephen D. Creech, NASA's assistant deputy associate administrator for the moon to Mars program, said in an announcement about the partnership. "For all of the money we are spending, NASA should NOT be talking about going to the Moon - We did that 50 years ago. They should be focused on the much bigger things we are doing, including Mars (of which the Moon is a part), Defense and Science!" Trump wrote in a post on X in 2019. Trump has said he would create a [[w:Space_National_Guard|Space National Guard]], an idea that lawmakers in Congress have been proposing since 2021. Critics agree that a focus on spaceflight could come at the expense of "Earth and atmospheric sciences at NASA and the [[w:National_Oceanic_and_Atmospheric_Administration|National Oceanic and Atmospheric Administration]] (NOAA), which have been cut during the Biden era."<ref>{{Cite web|url=https://www.newsweek.com/elon-musk-donald-trump-nasa-space-policy-1990599|title=Donald Trump and Elon Musk could radically reshape NASA. Here's how|last=Reporter|first=Martha McHardy US News|date=2024-11-27|website=Newsweek|language=en|access-date=2024-12-02}}</ref> Regarding his goal and SpaceX's corporate mission of colonising Mars, Mr. Musk has stated that "The DOGE is the only path to extending life beyond earth"{{sfn|Economist 11/23|2024}} == [[w:Federal_Aviation_Administration|Federal Aviation Administration]] == Musk has often complained about the FAA "smothering" innovation, boasting that he can build a rocket faster than the agency can process the "Kafkaesque paperwork" required to make the relevant approvals.{{sfn|Economist 11/23|2024}} == Department of Transportation == Mr. Trump's transition team told advisors that it plans to make a federal framework for self-driving cars. Mr. Trump had a call with Sundar Pichai and Mr. Musk. == Department of Education == [[w:United_States_Department_of_Education|w:Department of Education]] has 4,400 employees – the smallest staff of the Cabinet agencies<ref>{{Cite web|url=https://www2.ed.gov/about/overview/fed/role.html|title=Federal Role in Education|date=2021-06-15|website=www2.ed.gov|language=en|access-date=2022-04-28}}</ref> – and a 2024 budget of $238 billion.<ref name="DOE-mission">{{Cite web|url=https://www.usaspending.gov/agency/department-of-education?fy=2024|title=Agency Profile {{!}} U.S. Department of Education|website=www2.ed.gov|access-date=2024-11-14}}</ref> The 2023 Budget was $274 billion, which included funding for children with disabilities ([[wikipedia:Individuals_with_Disabilities_Education_Act|IDEA]]), pandemic recovery, early childhood education, [[wikipedia:Pell_Grant|Pell Grants]], [[wikipedia:Elementary_and_Secondary_Education_Act|Title I]], work assistance, among other programs. This budget was down from $637.7 billion in 2022.<ref>{{Cite web|url=https://www.future-ed.org/what-the-new-pisa-results-really-say-about-u-s-schools/|title=What the New PISA Results Really Say About U.S. Schools|date=2021-06-15|website=future-ed.com|language=en|access-date=2024-11-14}}</ref> The WSJ proposes that the Civil Rights division be absorbed into the Department of Justice, and that its outstanding loan portolio be handled by the Department of the Treasury. Despite the redundancies, its unlikely that it will be abolished, which would require congressional action and buy-in from Democrats in the Senate; Republicans don’t have enough votes to do it alone. A republican appointee is expected to push back against federal education overreach and progressive policies like DEI. <ref>[https://www.wsj.com/opinion/trump-can-teach-the-education-department-a-lesson-nominee-needs-boldness-back-school-choice-oppose-woke-indoctrination-ddf6a38d<nowiki> Trump can teach the Education Department a Lesson. WSJ 11/20/2024]</nowiki></ref> == Consumer Financial Protection Bureau == The [[w:Consumer_Financial_Protection_Bureau|Consumer Financial Protection Bureau]] (CFPB). Said Mr. Musk "Delete the CFPB. There are too many duplicative regulatory agencies"<ref name=":0">{{Cite news|url=https://www.wsj.com/politics/policy/elon-musk-doge-conflict-of-interest-b1202437?page=1|title=Musk’s DOGE Plans Rely on White House Budget Office. Conflicts Await.|last=Schwartz|first=Brian|work=The Wall Street Journal}}</ref> == U.S. Department of Health and Human Services == [[w:U.S._Department_of_Health_and_Human_Services|U.S. Department of Health and Human Services]] was authorized a budget for [[w:2020_United_States_federal_budget|fiscal year 2020]] of $1.293 trillion. It has 13 operating divisions, 10 of which constitute the [[w:United_States_Public_Health_Service|Public Health Services]], whose budget authorization is broken down as follows:<ref name="hhs_budget_fy2020">{{cite web|url=https://www.hhs.gov/about/budget/fy2020/index.html|title=HHS FY 2020 Budget in Brief|date=October 5, 2019|website=HHS Budget & Performance|publisher=United States Department of Health & Human Services|page=7|access-date=May 9, 2020}}</ref> {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Food and Drug Administration|Food and Drug Administration]] (FDA) |$3,329 MM |- |[[w:Health Resources and Services Administration|Health Resources and Services Administration]] (HRSA) |$11,004 |- |[[w:Indian Health Service|Indian Health Service]] (IHS) |$6,104 |- |[[w:Centers for Disease Control and Prevention|Centers for Disease Control and Prevention]] (CDC) |$6,767 |- |[[w:National Institutes of Health|National Institutes of Health]] (NIH) |$33,669 |- |[[w:Substance Abuse and Mental Health Services Administration|Substance Abuse and Mental Health Services Administration]] (SAMHSA) |$5,535 |- |[[w:Agency for Healthcare Research and Quality|Agency for Healthcare Research and Quality]] (AHRQ) |$0 |- |[[w:Centers for Medicare & Medicaid Services|Centers for Medicare & Medicaid Services]] (CMMS) |$1,169,091 |- |[[w:Administration for Children and Families|Administration for Children and Families]] (ACF) |$52,121 |- |[[w:Administration for Community Living|Administration for Community Living]] (ACL) |$1,997 |- |} {| class="wikitable sortable" !Program !2020 Budget !2024 Budget ! employees !2025 Budget !2026 Budget |- |[[w:Departmental Management|Departmental Management]] |$340 |- |Non-Recurring Expense Fund |$-400 |- |[[w:Office of Medicare Hearings and Appeals|Office of Medicare Hearings and Appeals]] |$186 |- |[[w:Office of the National Coordinator|Office of the National Coordinator]] |$43 |- |[[w:Office for Civil Rights|Office for Civil Rights]] |$30 |- |[[w:Office of Inspector General|Office of Inspector General]] |$82 |- |[[w:Public Health and Social Services Emergency Fund|Public Health and Social Services Emergency Fund]] |$2,667 |- |[[w:Program Support Center|Program Support Center]] |$749 |- |Offsetting Collections |$-629 |- |Other Collections |$-163 |- |'''TOTAL''' |'''$1,292,523''' |} The FY2020 budget included a $1.276 billion budget decrease for the Centers for Disease Control, and a $4.533 billion budget decrease for the National Institutes of Health. These budget cuts, along with other changes since 2019, comprised a total decrease of over $24 billion in revised discretionary budget authority across the entire Department of Health and Human Services for Fiscal Year 2020.<ref name="hhs_budget_fy2020" /> Additional details of the budgeted outlays, budget authority, and detailed budgets for other years, can be found at the HHS Budget website.<ref>{{cite web|url=http://WWW.HHS.GOV/BUDGET|title=Health and Human Services: Budget and Performance|publisher=United States Department of Health & Human Services|access-date=May 9, 2020}}</ref> [[w:Robert_F._Kennedy_Jr.|Robert F. Kennedy Jr.]] has been nominated as [[w:United_States_Secretary_of_Health_and_Human_Services|United_States_Secretary_of_Health_and_Human_Services]] He is an American politician, [[Environmental law|environmental lawyer]], [[anti-vaccine activist]], and anti-packaged food industry activist, anti-pharmaceutical industry activist, who will be nominated to serve as [[United States Secretary of Health and Human Services]],<ref name="v502">{{cite web|url=https://www.forbes.com/sites/saradorn/2024/11/14/rfk-jr-launches-independent-2024-run-here-are-all-the-conspiracies-he-promotes-from-vaccines-to-mass-shootings/|title=Trump Taps RFK Jr. As Secretary Of Health And Human Services: Here Are All The Conspiracies He's Promoted|last=Dorn|first=Sara|date=2024-11-14|website=Forbes|access-date=2024-11-15}}</ref> with the mission of "Making America Healthy Again". He is the chairman and founder of [[Children's Health Defense]], an anti-vaccine advocacy group and proponent of [[COVID-19 vaccine misinformation|dubious COVID-19 vaccine information]].<ref name="Smith_12/15/2021" /><ref name="KW" /> === National Institutes of Health === Nominated to lead the [[w:National_Institutes_of_Health|National Institutes of Health]] is [[w:Jay_Bhattacharya|Jay Bhattacharya]]. The NIH distributes grants of ~$50bn per year. Mr. Battacharya has announced the following priorities for funding: * cutting edge research, saying that the NIH has become "sclerotic". * studies aimed at replicating the results of earlier studies, to address the problem of scientific fraud or other factors contributing to the the [[w:Replication_crisis|replication crisis]] * refocusing on [[w:Chronic_diseases|chronic diseases]], whose research is underfunded, away from [[w:Infectious_diseases|infectious diseases]], whose research is overfunded. * ending [[w:Gain-of-function|gain-of-function]] research. Jay Bhattacharya wrote a March 25 2020 op-ed "Is the Coronavirus as Deadly as They Say?", with colleague Eran Bendavid, arguing that many asymptomatic cases of COVID-19 were going undetected. The hypothesis was confirmed in April 2020 when he and several colleagues published a study showing that Covid anti-bodies in Santa Clara county were 50 times the recorded infection rate. This implied, he said "a lower inflection mortality rate than public health authorities were pushing at a time when they and the media thought it was a virtue to panic the population".<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.wsj.com/opinion/the-man-who-fought-fauci-and-won-trump-nih-nominee-jay-bhattacharya-covid-cancel-culture-4a0650bd?page=1|title=The Man Who Fought Fauci - and Won|last=Varadarajan|first=Tunku|date=6 December 2024|work=WSJ}}<nowiki></ref></nowiki></ref> == CPB, PBS, NPR == [[w:Howard_Husock|Howard Husock]] suggest that instead of zeroing its $535 million budget, Republicans reform the [[w:Public_Broadcasting_Act|Public Broadcasting Act]] to eliminate bias and improve local journalism.<ref>https://www.wsj.com/opinion/the-conservative-case-for-public-broadcasting-media-policy-2d4c3c9f?page=1</ref> == Reform Entitlements == === Healthcare and Medicare === [[w:ObamaCare|ObamaCare]] started as a plausible scheme for universal, cost-effective health insurance with subsidies for the needy. Only the subsidies survive because the ObamaCare policies actually delivered are so overpriced nobody would buy them without a subsidy.<ref>[https://www.wsj.com/opinion/elons-real-trump-mission-protect-growth-department-of-government-efficiency-appointments-cabinet-9e7e62b2]</ref> See below: Department of Health and Human Services === Social Security === Even FDR was aware of its flaw: it discourages working and saving. === Other === Small-government advocate [[w:Ron_Paul|Ron Paul]] has suggested to cut aid to the following "biggest" welfare recipients: * The [[w:Military-industrial_complex|Military-industrial complex]] * The [[w:Pharmaceutical-industrial_complex|Pharmaceutical-industrial complex]] * The [[w:Federal_Reserve|Federal Reserve]] To which Mr. Musk replied, "Needs to be done".<ref>{{Cite web|url=https://thehill.com/video/ron-paul-vows-to-join-elon-musk-help-eliminate-government-waste-in-a-trump-admin/10191375/|title=Ron Paul vows to join Elon Musk, help eliminate government waste in a Trump admin|date=2024-11-05|website=The Hill|language=en-US|access-date=2024-12-09}}</ref> == Office of Management and Budget == The White House [[w:Office_of_Management_and_Budget|Office of Management and Budget]] (OMB) guides implementation of regulations and analyzes federal spending. Mssrs. Musk and Ramaswamy encouraged President-elect Trump to reappoint his first term director [[w:Russell_Vought|Russell Vought]], which he did on 22nd Nov.<ref name=":0" /> == Personnel == Transition spokesman [[w:Brian_Hughes|Brian Hughes]] said that "Elon Musk is a once-in-a-generation business leader and our federal bureaucracy will certainly benefit from his ideas and efficiency". About a dozen Musk allies have visited Mar-a-Lago to serve as unofficial advisors to the Trump 47 transition, influencing hiring at many influential government agencies.<ref><nowiki><ref></nowiki>{{Cite news|url=https://www.nytimes.com/2024/12/06/us/politics/trump-elon-musk-silicon-valley.html?searchResultPosition=1|title=The Silicon Valley Billionaires Steering Trump’s Transition|date=8 December 2024|work=NYT}}<nowiki></ref></nowiki></ref> Marc Andreesen has interviewed candidates for State, Pentagon, and DHHS, and has been active pushing for rollback of Biden's cryptocurrency regulations, and rollback of Lina Khan anti-trust efforts with the FTC, and calling for contracting reform in Defense dept. Jared Birchall has interviewed candidates for State, and has advised on Space police and has put together councils for AI and Cryptocurrency policy. David Sacks was named "White House AI and Crypto Czar" Shaun MacGuire has advised on picks for intelligence community and has interviewed candidates for Defense. Many tech executives are considering part-time roles advising the DOGE. Antonio Gracias and Steve Davis from Musk's "crisis team" have been active. John Hering Other Silicon Valley players who have advised Trump or interviewed candidates: * [[w:Larry_Ellison|Larry Ellison]] has sat in on Trump transition 47 meetings at Mar-a-Lago. * Mark Pincus * David Marcus * Barry Akis * Shervin Pishevar has called for privitization of the USPS, NASA, and the federal Bureau of Prisons. Called for creating an American sovereign wealth fund, and has said that DOGE "could lead a revolutionary restructuring of public institutions." == History and Miscellaneous facts == See also: [[w:Department_of_Government_Efficiency#History|Department of Government Efficiency — History]] DOGE's work will "conclude" no later than July 4, 2026, the 250th anniversary of the signing of the [[United States Declaration of Independence|U.S. Declaration of Independence]],<ref>{{Cite web|url=https://thehill.com/policy/4987402-trump-musk-advisory-group-spending/|title=Elon Musk, Vivek Ramaswamy to lead Trump's Department of Government Efficiency (DOGE)|last=Nazzaro|first=Miranda|date=November 13, 2024|website=The Hill|language=en-US|access-date=November 13, 2024}}</ref> also coinciding with America's [[United States Semiquincentennial|semiquincentennial]] celebrations and a proposed "Great American Fair". Despite its name it is not expected to be a [[wikipedia:United_States_federal_executive_departments|federal executive department]], but rather may operate under the [[Federal Advisory Committee Act]],<ref>{{Cite web|url=https://www.cbsnews.com/news/trump-department-of-government-efficiency-doge-elon-musk-ramaswamy/|title=What to know about Trump's Department of Government Efficiency, led by Elon Musk and Vivek Ramaswamy - CBS News|last=Picchi|first=Aimee|date=2024-11-14|website=www.cbsnews.com|language=en-US|access-date=2024-11-14}}</ref> so its formation is not expected to require approval from the [[wikipedia:United_States_Congress|U.S. Congress]]. NYT argues that records of its meetings must be made public.{{Cn}} As an advisor rather than a government employee, Mr. Musk will not be subject to various ethics rules.{{sfn|Economist 11/23}} Musk has stated that he believes such a commission could reduce the [[wikipedia:United_States_federal_budget|U.S. federal budget]] by $2 trillion, which would be a reduction of almost one third from its 2023 total. [[Maya MacGuineas]] of the [[Committee for a Responsible Federal Budget]] has said that this saving is "absolutely doable" over a period of 10 years, but it would be difficult to do in a single year "without compromising some of the fundamental objectives of the government that are widely agreed upon".<ref>{{Cite web|url=https://thehill.com/business/4966789-elon-musk-skepticism-2-trillion-spending-cuts/|title=Elon Musk draws skepticism with call for $2 trillion in spending cuts|last=Folley|first=Aris|date=2024-11-03|website=The Hill|language=en-US|access-date=2024-11-14}}</ref> [[wikipedia:Jamie_Dimon|Jamie Dimon]], the chief executive officer of [[wikipedia:JPMorgan_Chase|JPMorgan Chase]], has supported the idea. Some commentators questioned whether DOGE is a conflict of interest for Musk given that his companies are contractors to the federal government. The body is "unlikely to have any regulatory teeth on its own, but there's little doubt that it can have influence on the incoming administration and how it will determine its budgets".<ref>{{Cite web|url=https://www.vox.com/policy/384904/trumps-department-of-government-efficiency-sounds-like-a-joke-it-isnt|title=Trump tapped Musk to co-lead the "Department of Government Efficiency." What the heck is that?|last=Fayyad|first=Abdallah|date=2024-11-13|website=Vox|language=en-US|access-date=2024-11-14}}</ref> Elon Musk had called [[w:Federico_Sturzenegger|Federico Sturzenegger]], Argentina's [[Minister of Deregulation and Transformation of the State]] ([[w:es:Ministerio_de_Desregulación_y_Transformación_del_Estado|es]]), to discuss imitating his ministry's model.<ref>{{Cite web|url=https://www.infobae.com/economia/2024/11/08/milei-brindo-un-nuevo-apoyo-a-sturzenegger-y-afirmo-que-elon-musk-imitara-su-gestion-en-eeuu/|title=Milei brindó un nuevo apoyo a Sturzenegger y afirmó que Elon Musk imitará su gestión en EEUU|date=November 8, 2024|website=infobae|language=es-ES|access-date=November 13, 2024}}</ref> == Reception and Criticism == See also: [[w:Department_of_Government_Efficiency#Reception|w:Department of Government Efficiency — Reception]] The WSJ reports that Tesla's Texas facility dumped toxic wastewater into the public sewer system, into a lagoon, and into a local river, violated Texas environmental regulations, and fired an employee who attempted to comply with the law.{{Cn}} The Economist estimates that 10% of Mr. Musk's $360bn personal fortune is derived from contracts and benefits from the federal government, and 15% from the Chinese market.{{sfn|Economist 11/23}} == See also == * [[w:Second_presidential_transition_of_Donald_Trump|Second presidential transition of Donald Trump]] * [[w:United_States_federal_budget#Deficits_and_debt|United States federal budget - Deficits and debt]] * [[w:United_States_Bureau_of_Efficiency|United States Bureau of Efficiency]] – United States federal government bureau from 1916 to 1933 * [[w:Brownlow_Committee|Brownlow Committee]] – 1937 commission recommending United States federal government reforms * [[w:Grace_Commission|Grace Commission]] – Investigation to eliminate inefficiency in the United States federal government * [[w:Hoover_Commission|Hoover Commission]] – United States federal commission in 1947 advising on executive reform * [[w:Keep_Commission|Keep Commission]] * [[w:Project_on_National_Security_Reform|Project on National Security Reform]] == Notes == {{reflist}} == References == {{refbegin}} * {{Cite web|url=https://www.newsweek.com/bernie-sanders-finds-new-common-ground-elon-musk-1993820|title=Bernie Sanders finds new common ground with Elon Musk|last=Reporter|first=Mandy Taheri Weekend|date=2024-12-01|website=Newsweek|language=en|access-date=2024-12-02 |ref={{harvid|Newsweek 12/01|2024}} }} * {{Cite news|url=https://www.economist.com/briefing/2024/11/21/elon-musk-and-donald-trump-seem-besotted-where-is-their-bromance-headed|title=Elon Musk and Donald Trump seem besotted. Where is their bromance headed?|work=The Economist|access-date=2024-12-04|issn=0013-0613 |ref={{harvid|Economist 11/23|2024}} }} <references group="lower-alpha" /> {{refend}} jw8cvu5qdu90jvq7ot7xilocbpjke8g 0 316907 2690983 2690064 2024-12-09T07:22:07Z Bert Niehaus 2387134 /* Web links */ 2690983 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] 'curvus'' 'bent, curved') is a [[w:en:Dimension (mathematics)|eindimensionales]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space15105. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== {{Commonscat|Curves|Curves}} {{Wiktionary|Curve}} == Page Information == === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> <noinclude>[[de:Kurs:Funktionentheorie/Kurven]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> p1xddhtte26s2j890i2k97x7ywtjpcs 2690997 2690983 2024-12-09T10:59:48Z Bert Niehaus 2387134 /* Page Information */ 2690997 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] 'curvus'' 'bent, curved') is a [[w:en:Dimension (mathematics)|eindimensionales]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space15105. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== {{Commonscat|Curves|Curves}} {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Kurven]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> km95v0o1nqc5iv9bj9v7epgxht2tw4y 2690998 2690997 2024-12-09T11:01:29Z Bert Niehaus 2387134 /* Web links */ 2690998 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] 'curvus'' 'bent, curved') is a [[w:en:Dimension (mathematics)|eindimensionales]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space15105. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== * Images on Wikicommons about Curves - {{Commonscat|Curves|Curves}} * Wiktionary Definition - {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Kurven]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> 72yawjibev5ky0blfzv6niprd28le9q 2690999 2690998 2024-12-09T11:03:07Z Bert Niehaus 2387134 /* Web links */ 2690999 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] 'curvus'' 'bent, curved') is a [[w:en:Dimension (mathematics)|eindimensionales]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space15105. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== * Images on Wikicommons about Curves {{Commonscat|Curves|Curves}} * Wiktionary - Dictionary entry for Curve {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Kurven]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> 3xmvbby148ju3oh33vuv1opw7yai81j 2691000 2690999 2024-12-09T11:04:52Z Bert Niehaus 2387134 /* Introduction */ 2691000 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] ''curvus'' for "bent", "curved") is a [[w:en:Dimension (mathematics)|eindimensionales]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== * Images on Wikicommons about Curves {{Commonscat|Curves|Curves}} * Wiktionary - Dictionary entry for Curve {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude>[[de:Kurs:Funktionentheorie/Kurven]]</noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> 551gxet4su30jys4xrdfr9w3ej4unsd 2691001 2691000 2024-12-09T11:39:54Z 139.14.134.44 /* Introduction */ 2691001 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] ''curvus'' for "bent", "curved") is a [[w:en:Dimension (mathematics)|one dimensionals]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:w:en:w:en:Polynomialialialial|w:en:w:en:Polynomialialialial]], the curve is called ''[[w:en:Algebraic curve|algebraisch]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== * Images on Wikicommons about Curves {{Commonscat|Curves|Curves}} * Wiktionary - Dictionary entry for Curve {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude> [[de:Kurs:Funktionentheorie/Kurven]] </noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> c0leompcq8xptdskx9il7udddd0jje9 2691002 2691001 2024-12-09T11:58:40Z Eshaa2024 2993595 2691002 wikitext text/x-wiki ==Introduction== In the [[w:en:Mathematics|Mathematics]] a ''curve'' (of [[w:en:Latin|lat.]] ''curvus'' for "bent", "curved") is a [[w:en:Dimension (mathematics)|one dimensionals]] [[w:en:Mathematical object|object]] in a two-dimensional plane (i.e. a curve in the plane) or in a higher-dimensional space. ==Parameter representations== * Multidimensional analysis: A continuous mapping <math display="inline"> f:[a,b]\to \mathbb{R}^n </math> is a curve in the <math display="inline"> \mathbb{R}^n </math>. * Complex Analysis: Continuous mapping <math display="inline"> f:[a,b] \to \mathbb{C} </math> is a path in <math display="inline"> \mathbb{C} </math> (see also [[w:en:Path_(topology)|path for integration]]). ==Explanatory notes== A curve/a way is a mapping. It is necessary to distinguish the track of the path or the [[w:en:Image (mathematics)|image]] of a path from the mapping graph. A path is a steady mapping of a [[w:en:Interval (mathematics)|interval]] in the space considered (e.g. <math display="inline"> \mathbb{R}^n </math> or <math display="inline"> \mathbb{C} </math>). ===Example 1 - Plot=== [[File:Cubic_with_double_point.svg|Cubic_with_double_point.svg]] <math display="inline"> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2, </math> <math display="inline"> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big) </math> ===EXAMPLE 1 Curve as a solution of an algebraic equation=== <table> <tr> <td> [[File:Cubic with double point.svg|150px|Cubic with double point]] </td> <td vslign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> resp. <math>y^2 = x^2 (x+1)</math>. Determine for the curve all <math display="inline"> (t_1,t_2) \in \mathbb{R}^2 </math> with <math display="inline"> \gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2 </math> </td></tr></table> ===Examples 2=== The mapping * <math display="inline"> \widetilde{\gamma_2} \colon [ 0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit circle|Unit circle]] in the plane <math display="inline"> \mathbb{R}^2 </math>. * <math display="inline"> \gamma_2 \colon [ 0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit circle|Unit circle]] in the Gaussian number level <math display="inline"> \mathbb{C} </math>. ===Examples 3=== The mapping : <math display="inline"> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2,\quad t\mapsto \gamma_3(t) =\big(t^2-1,t(t^2-1)\big) </math> describes a curve with a simple double point at <math display="inline"> (0,0) </math>, corresponding to the parameter values <math display="inline"> t=1 </math> and <math display="inline"> t=-1 </math>. ===Direction=== As a result of the parameter representation, the curve receives a ''directional direction'' in the direction of increasing parameter.<ref name="CITE1">H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, lSBN 978-3-642-97840-1, 23.5</ref><ref name="cite2">H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematics. 12. Auflage. Walter de Gruyter, 1994, lSBN 978-3-486-78544-9 </ref> ==Curve as Image of Path== Let <math display="inline"> \gamma:[a,b] \to \mathbb{C} </math> or <math display="inline"> \gamma:[a,b] \to \mathbb{R}^n </math> be a path. is the image of a path : <math display="inline"> Spur ( \gamma ) := \left\{ (t,\gamma(t)) \ | \ a \leq t \leq b \right\} </math>. === Difference - Graph und Curve === For a curve <math>\gamma:[a,b] \to \mathbb{R}^2</math> the Supr or curve is a subset of <math> \mathbb{R}^2 </math>, while the graph of function <math> Graph(\gamma) \subset \mathbb{R}^3 </math> is. === Task - Plot Graph und Curve === use [[CAS4Wiki]] : :<math> \begin{array}{rrcl} \gamma: & [0,6\pi] & \rightarrow & \mathbb{R}^2 \\ & t & \mapsto & \gamma(t) = \left( 3\cdot \cos(t), \sin(t) \right) \end{array} </math> ===Animation of the track=== [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Abrollkurve]] ==Curves in Geogebra== First create a slider for the variable <math display="inline"> t\in [0,2\pi] </math> and two points <math display="inline"> K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2 </math> or <math display="inline"> K_2=(\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> and generate with <math display="inline"> K := K_1 + K_2 </math> the sum of both location vectors of <math display="inline"> K_1 </math> and <math display="inline"> K_2 </math>. Analyze the parameterization of the curves. Geogebra: <pre> K_1:(2*cos(t),2 * sin(t)) K_2: (cos(3*t),sin(3*t)) </code>, <pre><code> K: K_1+K_2 </pre> :<math display="block"> \gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interaktive Exampe in Geogebra] ==Equation representations== A curve can also be described by one or more equations in the coordinates. The solution of the equations represents the curve: * The equation <math display="inline"> x^2+y^2=1 </math> describes the unit circle in the plane. * The equation <math display="inline"> y^2=x^2(x+1) </math> describes the curve indicated above in parameter representation with double point. If the equation is given by a [[w:en:Polynomial]], the curve is called ''[[w:en:Algebraic curve|algebraic]]'. == Graph of a function== [[w:en:Graph of a function|graph of a functionen]] are a special case of the two forms indicated above: The graph of a function : <math display="inline"> f \colon D\to\mathbb{R},\quad x\mapsto f(x) </math> can be either as a parameter representation <math display="inline"> \gamma \colon D \to \mathbb R^2,\quad t\mapsto(t,f(t)) \} </math> or as equation <math display="inline"> y=f(x) </math>, wherein the solution quantity of the equation represents the curve by <math display="inline"> \{ (x,y) \in \mathbb R^2\mid y=f(x) \} </math>. If the [[w:en:Mathematics education|Mathematics Education]] of [[w:en:w:en:Curve sketching|w:en:Curve sketching]] is spoken, this special case is usually only said. ==Closed curves== Closed curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> are continuous mappings with <math display="inline"> \gamma(a) = \gamma(b) </math>. In the function theory, we need curves <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> in <math display="inline"> \mathbb C </math>, which can be continuously differentiated. These are called integration paths. ===Number of circulations in the complex numbers=== Smooth closed curves can be assigned a further number, the [[w:en:Winding number|Umlaufzahl]], which curve is parameterized according to the arc curve <math display="inline"> \gamma \colon [a,b]\to\mathbb C </math> by :<math display="block"> \mu(\gamma,z):= \frac{1}{2\pi i}\int_\gamma \frac{1}{\xi - z}\,d\xi := \int_a^b \frac{1}{\gamma(t) - z}\cdot \gamma'(t)\,dt </math> is given. The [[w:en:Umlaufsatz|Umlaufsatz]] analogously to a curve in <math display="inline"> R^2 </math>, states that a simple closed curve has the number of revolutions <math display="inline"> 1 </math> or <math display="inline"> -1 </math>. ==Curves as independent object== Curves without surrounding space are relatively uninteresting in the [[w:en:Differential geometry|Differential geometry]] because each one-dimensional [[w:en:Manifold|Manifold]] [[w:en:w:en:Diffeomorphism|diffeomorph]] for real straight lines <math display="inline"> \mathbb R </math> or for the unit circle line <math display="inline"> S^1 </math>. Also properties such as the [[w:en:Curvature|Curvature]] of a curve cannot be determined intrinsically. In the [[w:en:Algebraic geometry|algebraischen Geometrie]] and associated in the [[w:en:Complex Analysis|komplexen Analysis]], “curves” are generally understood as one-dimensional [[w:en:w:en:complex manifold|w:en:complex manifold]]en, often also referred to as [[w:en:Riemann surface|Riemann surface]]. These curves are independent study objects, the most prominent example being the [[w:en:Elliptic curve|elliptischen Kurven]]. ''See' [[w:en:Algebraic geometry|Curve (algebraic geometry)]] ==Historical== The first book of [[w:en:w:en:Euclid's Elements|Elemente]] of [[w:en:Euclid|Euclid]] began with the definition : ''A point is what has no parts. A curve is a length without width.'' This definition can no longer be maintained today, as there are, for example, [[w:en:Peano curve|Peano-curven]], i.e., continuous [[w:en:surjective|surjectivee]] mappings <math display="inline"> f\colon\mathbb{R}\to \mathbb{R}^2 </math>, which fill the entire plane <math display="inline"> \mathbb R^2 </math>. On the other hand, it follows from [[w:en:Sard's theorem|Sard's theorem]] that each differentiable curve has the area content zero, i.e. actually as required by Euclid 'no width'. ==Interactive display of curves in geogebra== * [https://www.geogebra.org/m/e3hhdrvq Tangent vector] in <math display="inline"> \mathbb{R}^2 </math> for a [[w:en:Path (topology)|path]] <math display="inline"> \gamma:[a,b]\to \mathbb{R}^2 </math> with a tangent vector <math display="inline"> \gamma'(t)\in \mathbb{R}^2 </math> defined by the derivation <math display="inline"> \gamma':[a,b]\to \mathbb{R}^2 </math> * [https://www.geogebra.org/m/srmgcsZX Curves created by reflector on wheels of a bicycle] as an example of curves - [[w:en:Cycloid|Cycloid]] * [https://www.geogebra.org/m/ppuvs3ge Example of Roulette curves] - see also [[w:en:Roulette (curve)|Roulette (curve)]] ==See also== * [[w:en:Curve|curve in a vector space]] in <math display="inline"> \mathbb{R}^3 </math> * [[w:en:Curvature|Curvature]] * [[w:en:Cycloid|Cycloid]] ==Literature== * Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry'. Birkhäuser, Boston 1997. * Wilhelm Klingenberg: ''A Course in Differential Geometry'. Springer, New York 1978. ==Individual evidence== <references /> ==Web links== * Images on Wikicommons about Curves {{Commonscat|Curves|Curves}} * Wiktionary - Dictionary entry for Curve {{Wiktionary|Curve}} == Page Information == You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' === Wiki2Reveal === The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator]. <!-- * Contents of the page are based on: ** [https://en.wikipedia.org/wiki/Complex%20Analysis/Curves https://en.wikiversity.org/wiki/Complex%20Analysis/Curves] --> * [https://en.wikiversity.org/wiki/Complex%20Analysis/Curves This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type. * Source: Wikiversity https://en.wikiversity.org/wiki/Complex%20Analysis/Curves * see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Complex%20Analysis/Curves&author=Complex%20Analysis&language=en&audioslide=yes&shorttitle=Curves&coursetitle=Complex%20Analysis Wiki2Reveal]. <!-- * Next contents of the course are [[]] -->; === Translation and Version Control === This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity: * Source: [[v:de:Kurs:Funktionentheorie/Kurven|Kurs:Funktionentheorie/Kurven]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Kurven * Date: 12/2/2024 * [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity <span type="translate" src="Kurs:Funktionentheorie/Kurven" srclang="de" date="12/2/2024" time="17:45" status="inprogress"></span> [[Category:Wiki2Reveal]] <noinclude> [[de:Kurs:Funktionentheorie/Kurven]] </noinclude> <!-- <noinclude>[[en:Complex Analysis/Curves]]</noinclude> --> 8wi0awgc3z094yi4hgawfhg60rajnnc 0 316944 2690878 2690467 2024-12-08T16:18:36Z UniBambergIslamicStudies 2987517 2690878 wikitext text/x-wiki [[The Bamberg Introduction to the History of Islam (BIHI) 02|2 <<<]] — [[The Bamberg Introduction to the History of Islam (BIHI) 04|>>> 4]] = 3. The Prophet of Yathrib and the New Polity (622-630) = The center of the new religion shifts to the oasis of Yathrib, with warfare taking center stage. Muḥammad and his followers engage in battles against pagan Mecca and increasingly come into conflict with the Jews of Yathrib, who are ultimately expelled from the oasis. As the leader of the nascent community, Muḥammad implements a series of legal, social, and ritual reforms. == 3.1. Maghāzī – The Military Expeditions of Muḥammad == === 3.1.1. The Provocation of the Quraysh === Arab sources consistently report that Muḥammad arrived at the oasis of Yathrib on September 24, 622, following his emigration from Mecca. Having been expelled from his hometown, he considered it justifiable to engage in conflict against his former hometown. This is clearly reflected in two Qur'anic verses, widely recognized as the earliest revelations on the subject of warfare: {{quote|Sanction is given unto those who fight because they have been wronged; and Allah is indeed able to give them victory; Those who have been driven from their homes unjustly only because they said: Our Lord is Allah. [https://corpuscoranicum.de/en/verse-navigator/sura/22/verse/39/print – Q 22:39f]}} The war with Mecca, which Muḥammad waged from his new base in Yathrib, began with minor pinpricks. According to the chronology of [[w:Al-Waqidi|al-Wāqidī]], who composed a detailed account of Muḥammad's military expeditions ([https://de.wikipedia.org/wiki/Magh%C4%81z%C4%AB maghāzī]) in the early 9th century, Muḥammad dispatched his uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]] with a group of warriors seven months after his arrival in Yathrib to intercept a Meccan trade caravan returning from Syria under the leadership of [[w:Amr_ibn_Hisham|Abū Jahl]]. However, no combat occurred because a man from the [[w:Juhaynah|Juhaynah]] tribe, allied with both sides, intervened. During a [[w:Expedition_of_Ubaydah_ibn_al-Harith|second expedition]] in April 623, "the first arrow of Islam" was launched. The conflict with the Meccans soon disregarded traditional Arab religious norms, such as the obligation to maintain peace during the sacred months (see above [[The Bamberg Introduction to the History of Islam (BIHI) 01#1.3.3. Ancient Arabian Paganism and the Sacred Sites of Mecca|1.3.3.]]). For example, a unit commissioned by Muḥammad [[w:Raid_on_Nakhla|raided]] a Meccan caravan during the sacred month of [[w:Rajab|Rajab]] near [[w:Nakhla_(Saudi_Arabia)|Nakhla]], south of Mecca. According to tradition, this event prompted the following revelation: {{quote|They question [you] (O Muhammad) with regard to warfare in the sacred month. Say: Warfare therein is a great (transgression), but to turn (men) from the way of Allah, and to disbelieve in Him and in the Inviolable Place of Worship, and to expel His people thence, is a greater with Allah; for persecution is worse than killing. And they will not cease from fighting against you till they have made you renegades from your religion, if they can. [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/217/print – Q 2:217]}} <!-- linked German article for Maghāzī as there isn't an English one. -linked Raid on Nakhla, Nakhla_(Saudi_Arabia), and Expedition_of_Ubaydah_ibn_al-Harith articles --> From this Qur'anic verse, it is evident that the continued existence of the old religion in Mecca posed a constant temptation for Muḥammad’s followers to abandon their faith. Since many of them apparently found military combat (''qitāl'') undesirable, Muḥammad now declared it a duty (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/216/print 2:216]) and elevated it to a religious level by designating it as ''jihād fī sabīl Allāh'' (“striving [[w:Fi_sabilillah|in the way of God]]”, as stated in the subsequent verse Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/218/print 2:218]). This term has also been adopted into the English language in the form of [[w:Jihad|Jihad]]. [[File:Balami - Tarikhnama - The Battle of Badr - The death of Abu Jahl, and the casting of the Meccan dead into dry wells (cropped).jpg|thumb|Illustration of the [[w:Battle_of_Badr|Battle of Badr]] in a Persian manuscript, early 14th century]] The first major confrontation between the Meccans and Muḥammad’s followers took place in March 624 near the site of [[w:Badr,_Saudi_Arabia|Badr]], approximately 130 kilometers southwest of Yathrib. Muḥammad had received information about a wealthy Meccan caravan returning from Syria. With 300 men, including members of the [[w:Banu_Muzaina|Muzaynah]] tribe allied with the [[w:Banu_Aws|Aws]], he set out for Badr, situated along the coastal road, to intercept the caravan. A battle ensued between Muḥammad's forces and a Meccan army of approximately 950 men, which had rushed to the caravan's aid under the command of Muḥammad’s bitter adversary Abū Jahl. Muḥammad's forces achieved an unexpected victory. The Meccans suffered between 45 and 70 fatalities, with a similar number taken prisoner. Among the fallen Meccans were several prominent figures, including Abū Jahl. In contrast, Muḥammad’s followers lost only 14 men and captured substantial spoils of war. <!-- linked city of Badr --> Following the battle, Muḥammad had some of the prisoners beheaded, including his former adversary [[w:Nadr_ibn_al-Harith|al-Naḍr ibn al-Ḥārith]]. The [[w:Battle_of_Badr|victory at Badr]] was of immense military and religious significance for Muḥammad's followers. Apparently, however, not all of them contributed to this victory. This is evident from verses revealed after Badr, which clarify that those among the believers who “sit still” at home without a valid excuse are not equal in rank before God to the Mujāhidūn – those who engage in jihad (strive in the way of Allah) with their wealth and their lives (cf. Q [https://corpuscoranicum.de/en/verse-navigator/sura/4/verse/95/print 4:95f]). <!-- -ließ Muhammad einige der Gefangenen enthaupten: "ordered the execution of several prisoners" would sound more academic and formal in English, but I translated it as "had some of the prisoners beheaded" to align with German wording. -Pickthall (and others similarly) translate the verse as “those who strive in the way of Allah with their wealth and lives”. I translated it as 'engage in jihad' to align with the German but put 'strive in the way of Allah' in parentheses. --> === 3.1.2. The Defense Against the Meccan Counterattack === The defeat at Badr dealt a severe blow to the Quraysh of Mecca. They had long been regarded as one of the most powerful tribes in Arabia, and to some extent, their commercial success relied on this reputation. Their trade depended on cooperation with many other tribes, and now, insubordination from some of these tribes was to be anticipated. It was therefore of critical importance for the Quraysh to demonstrate that they still possessed the strength to exact revenge for the wrongs they had suffered. Ten weeks after the Battle of Badr, [[w:Abu_Sufyan_ibn_Harb|Abū Sufyān ibn Ḥarb]], who had assumed leadership of Mecca following the battle, carried out a swift raid on Yathrib. After setting fire to two houses, however, he quickly withdrew. [[File:The Prophet Muhammad and the Muslim Army at the Battle of Uhud, from the Siyer-i Nebi, 1595.jpg|thumb|A depiction of the Battle of Uhud in a [[w:Siyer-i_Nebi|Siyer-i Nebi]] from 1594, now part of the David Collection in Copenhagen.]] In the months that followed, Abū Sufyān succeeded in recruiting 3,000 well-equipped warriors. In March 625, he advanced toward Yathrib with this force, penetrating the oasis from its northwestern corner. At Mount Uhud, a battle ensued, with the momentum shifting back and forth between the two sides for a long time. As the tide of war began to shift in favor of Muḥammad’s followers, they started gathering the spoils. This prompted a group of Muḥammad’s archers to abandon their positions to turn their attention to the spoils. On the Meccan side, [[w:Khalid_ibn_al-Walid|Khālid ibn al-Walīd]], a prominent warrior, exploited the situation to sow confusion among the ranks of Muḥammad's followers and ultimately overpower them. However, in the end, Muḥammad’s followers succeeded in regaining critical positions, causing the Meccans to withdraw without permanently eliminating their adversary, Muḥammad. For Muḥammad’s followers, the [[w:Battle_of_Uhud|Battle of Uhud]] was nevertheless a bitter disappointment: not only because they had lost 50 to 70 men, including Muḥammad’s uncle [[w:Hamza_ibn_Abd_al-Muttalib|Ḥamza]], and Muḥammad himself had been injured, but also because they came to realize that divine support was not as assured as it had seemed after their victory at Badr. Several Qur’anic verses from this period affirm that those who are killed “in the way of God” are not truly dead but living (Q [https://corpuscoranicum.de/en/verse-navigator/sura/2/verse/154/print 2:154]), are provided for by their Lord (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/169/print 3:169]), have their sins forgiven (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/157/print 3:157]), and are admitted directly into Paradise (Q [https://corpuscoranicum.de/en/verse-navigator/sura/3/verse/195/print 3:195]). <!-- Q 3:165-168: is said to deal with the reason as to why they lost at Uhud. --> The conflict between Muḥammad and the Meccans was by no means concluded with the Battle of Uhud. As Muḥammad continued to disrupt Meccan trade and found an increasing number of allies among the Arabian Bedouins, the Meccans felt compelled to take action against him once more. In turn, they sought to recruit a number of Bedouin tribes to their side. These alliances demonstrate that the conflict between Mecca and Yathrib had by then extended to the surrounding regions of both cities. In July 625, the [[w:Banu_Sulaym|Banū Sulaym]], a tribe allied with the Quraysh, massacred a large number of Muslims at [[w:Massacre_of_Bi%27r_Ma%27una|Biʾr Maʿūna]], located between Mecca and Yathrib. In response, Muḥammad is said to have cursed the Banū Sulaym for an entire month. This practice has been preserved in a modified form as part of the [[w:Qunut|Qunūt]], a supplication recited during the morning prayer or the nightly [[w:Witr|Witr]] prayer. <!-- linked the article on the massacre & Witr prayer -the first paragraph mentions the wrongs Quraysh has suffered, whereas this paragraph shows the ongoing reasons as to why the Quraysh felt they had to attack. Can the Medinan side also be described? My understanding is that among other things Quraysh continued to oppress Muslims who had not migrated to Medina. There are documented cases of torture and economic deprivations. Quraysh also would cut off trade relationships with tribes that supported the Muslim community and use families of migrants as leverage, no? --> At the beginning of 627, the Meccans and their allies advanced to Yathrib with a force of 10,000 men. Muḥammad, however, had a trench (''khandaq'') excavated around the less fortified areas of the oasis settlement, making it wide enough that a horse could not leap across. This move took the Meccans by such surprise that they were unable to devise an effective strategic response. What had been intended as an assault instead turned into a siege. Due to intrigues, however, the Meccan alliance collapsed after only 14 days, forcing an end to the [[w:Battle_of_the_Trench|siege of Yathrib]]. The Meccans ultimately withdrew without having achieved anything. 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This essay argues that the reproduction of a book's table of contents on wikiversity is fair use. * [[Wikiversity:Uploading files#Fair use considerations]] * [[Wikiversity:Academic freedom]] * [[Wikiversity:Copyrights]] * [[Wikiversity:Copyright issues]] * [[Meta:Avoid copyright paranoia]] * The article on [[w:Harry_Potter_and_the_Philosopher's_Stone#Podcast_version|Harry Potter and the Philosopher's Stone]], a fictional book, lists its table of contents. TODO * Adler's is a [[w:Wikipedia:FAQ/Copyright#Derivative_works]] of public domain. * Uncertain whether his "second table of contents" is [[w:Free_content|free content]] or [[w:Wikipedia:Non-free_content|non-free content]]. * If non-free, it may be permissible as "fair use" [[w:Wikipedia:FAQ/Copyright#What_is_fair_use?]] * [[w:Wikipedia:Copyrights#Using_copyrighted_work_from_others]] * [[w:Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing|Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing]] * [[w:User:Jaredscribe/WV:Toc_is_fair_use|User:Jaredscribe/WV:Toc is fair use]] If non-free, and if under some legal theory this is not allowed under fair use, then requesting the publisher: * [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] * [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] * [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. mxzyzv1uh7m8sl48jedrl9d4qt6xhmq 2690976 2690975 2024-12-09T05:39:42Z Jaredscribe 2906761 2690976 wikitext text/x-wiki {{policy proposal}} A "table of contents" is a "selection and arrangement" (and as such may be copyrighted). This essay argues that the reproduction of a book's table of contents on wikiversity is fair use. * [[Wikiversity:Uploading files#Fair use considerations]] * [[Wikiversity:Academic freedom]] * [[Wikiversity:Copyrights]] * [[Wikiversity:Copyright issues]] * [[Meta:Avoid copyright paranoia]] * The article on [[w:Harry_Potter_and_the_Philosopher's_Stone#Podcast_version|Harry Potter and the Philosopher's Stone]], a fictional book, lists its table of contents. * [[Wikipedia:FAQ/Copyright#What_is_fair_use?]] * Adler's is a [[w:Wikipedia:FAQ/Copyright#Derivative_works]] of public domain. * Uncertain whether his "second table of contents" is [[w:Free_content|free content]] or [[w:Wikipedia:Non-free_content|non-free content]]. * If non-free, it may be permissible as "fair use" [[w:Wikipedia:FAQ/Copyright#What_is_fair_use?]] * [[w:Wikipedia:Copyrights#Using_copyrighted_work_from_others]] * [[w:Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing|Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing]] * [[w:User:Jaredscribe/WV:Toc_is_fair_use|User:Jaredscribe/WV:Toc is fair use]] If non-free, and if under some legal theory this is not allowed under fair use, then requesting the publisher: * [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] * [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] * [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. 1u0zj6nic33qyilg49938h7yu71ml2y 2690977 2690976 2024-12-09T05:43:57Z Jaredscribe 2906761 2690977 wikitext text/x-wiki {{policy proposal}} A "table of contents" is a "selection and arrangement" (and as such may be copyrighted). This essay argues that the reproduction of a book's table of contents on wikiversity is fair use. * [[Wikiversity:Uploading files#Fair use considerations]] * [[Wikiversity:Academic freedom]] * [[Wikiversity:Copyrights]] * [[Wikiversity:Copyright issues]] * [[Meta:Avoid copyright paranoia]] * The article on [[w:Harry_Potter_and_the_Philosopher's_Stone#Podcast_version|Harry Potter and the Philosopher's Stone]], a fictional book, lists its table of contents. * Adler's is a [[w:Wikipedia:FAQ/Copyright#Derivative_works]] of public domain. * Uncertain whether his "second table of contents" is [[w:Free_content|free content]] or [[w:Wikipedia:Non-free_content|non-free content]]. * If non-free, it may be permissible as "fair use" [[w:Wikipedia:FAQ/Copyright#What_is_fair_use?]] * [[w:Wikipedia:Copyrights#Using_copyrighted_work_from_others]] * [[w:Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing|Wikipedia:Close_paraphrasing#When_there_are_a_limited_number_of_ways_to_say_the_same_thing]] * [[w:User:Jaredscribe/WV:Toc_is_fair_use|User:Jaredscribe/WV:Toc is fair use]] *[[Wikipedia:FAQ/Copyright#What_is_fair_use?]] : Under US copyright law, the primary things to consider when asking if something is fair use (set forth in Title 17, Chapter 1, Section 107) are: :# The purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes; :# The nature of the copyrighted work; :# The amount and substantiality of the portion used in relation to the copyrighted work as a whole; and :# The effect of the use upon the potential market for or value of the copyrighted work. : Asking yourself these questions might help you determine if something is fair use: :# ''Is it a for profit competitor or not? 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Did they intend to or were they trying to make the work widely republished (as with a press release)? Are you making it easy to find and buy the work if a viewer is interested in doing so?'' If non-free, and if under some legal theory this is not allowed under fair use, then requesting the publisher: * [[w:Wikipedia:Requesting_copyright_permission#For_text|Wikipedia:Requesting_copyright_permission#For_text]] * [[w:Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter|Wikipedia:Example_requests_for_permission#Generalized_Formal_Letter]] * [[w:Wikipedia:Declaration_of_consent_for_all_enquiries|Wikipedia:Declaration_of_consent_for_all_enquiries]]. h93d9lj4y6m6ofl449j9cr8763022ox 0 317056 2690860 2690812 2024-12-08T14:30:34Z Atcovi 276019 {{Prod}}; very closely a duplicate page with [[Complex Analysis/Curves]], except the latter is mixed with German 2690860 wikitext text/x-wiki {{Prod}} == Introduction == In [[w:en:Mathematics|mathematics]], a '''curve''' (from [[w:en:Latin|Latin]] curvus meaning "bent, curved") is a [[w:en:Dimension_(mathematics)|one-dimensional]] [[w:en:Mathematical_object|object]] in a two-dimensional plane (i.e., a plane curve) or in a higher-dimensional space (see [[w:en:Space_curve|space curve]]). == Parametric Representations == Multidimensional analysis: A continuous mapping <math>f:[a,b]\to \mathbb{R}^n</math> is a curve in <math>\mathbb{R}^n</math>. Complex analysis: A continuous mapping <math>f:[a,b]\to \mathbb{C}</math> is a path in <math>\mathbb{C}</math> (see also [[w:en:Path_(mathematics)|integration path]]). == Explanations == A curve/path is a mapping. It is essential to distinguish between the trace of the path, i.e., the [[w:en:Image_(mathematics)|image]] of a path, and the graph of the mapping. A path (contrary to colloquial use) is a continuous mapping from an [[w:en:Interval_(mathematics)|interval]] into the space considered (e.g., <math>\mathbb{R}^n</math> or <math>\mathbb{C}</math>). === Example 1 - Plot === [[File:Cubic with double point.svg|250px|]] <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math> === Example 1 - Curve as Solution Set of an Equation === <table> <tr> <td> [[File:Cubic with double point.svg|150px|]] </td> <td valign="top"> <math> \gamma_{1} \colon \mathbb{R} \to \mathbb{R}^2,</math> <math> t\mapsto \gamma_1(t) =\big(t^2-1,t(t^2-1)\big)</math>, or <math>y^2 = x^2 (x+1)</math>. Determine all <math>(t_1,t_2) \in \mathbb{R}^2</math> such that <math>\gamma_1(t_1)=\gamma_1(t_2) \in \mathbb{R}^2</math>. </td> </tr> </table> ==Example 2== The mapping <math>\widetilde{\gamma_2} \colon [0,2\pi ) \to \mathbb{R}^2,\quad t\mapsto \widetilde{\gamma_2}(t) =(\cos t,\sin t) </math> describes the [[w:en:Unit_circle|unit circle]] in the plane <math>\mathbb{R}^2</math>. <math>\gamma_2 \colon [0,2\pi ) \to \mathbb{C},\quad t\mapsto \gamma_2(t) = \cos (t) + i\sin (t) </math> describes the [[w:en:Unit_circle|unit circle]] in the Gaussian complex plane <math>\mathbb{C}</math>. ==Example 3== The figure: <math> \gamma_{3} \colon \mathbb{R} \to \mathbb{R}^2, \quad t \mapsto \gamma_3(t) = \big(t^2 - 1, t(t^2 - 1)\big) </math> describes a curve with a simple double point at <math>(0,0)</math>, corresponding to the parameter values <math>t=1</math> and <math>t=-1</math>. Direction Sense Through the parametric representation, the curve gains a '''direction''' in the direction of the increasing parameter. <ref>H. Neunzert, W.G. Eschmann, A. Blickensdörfer-Ehlers, K. Schelkes: Analysis 2: Mit einer Einführung in die Vektor- und Matrizenrechnung. Ein Lehr- und Arbeitsbuch. 2. Auflage. Springer, 2013, ISBN 978-3-642-97840-1, 23.5</ref> <ref>H. Wörle, H.-J. Rumpf, J. Erven: Taschenbuch der Mathematik. 12. Auflage. Walter de Gruyter, 1994, ISBN 978-3-486-78544-9 </ref> == Curve of a Path as Image Set== The curve of a path <math>\gamma:[a,b] \to \mathbb{C}</math> or <math>\gamma:[a,b] \to \mathbb{R}^n</math> is the [[w:en:Image_(mathematics)|image]] of a path: :<math> \mathcal{Crv}(\gamma) := \left\{ \gamma(t) \, | \, a \leq t \leq b \right\} </math> while [[w:de:Graph of function|graph of a function]] is the set :<math> \mathcal{G} (\gamma) := \left\{ (t,\gamma(t)) \, | \, a \leq t \leq b \right\}. </math> === Remark - Curve and Graph of a Path === Regarding <math>t</math> as point in time and <math> \gamma </math> as a movement of an object in a topological space. then we can describe the difference between curve and graph in the following way. * with the graph <math> \mathcal{G} (\gamma)</math> it is possible to identify at which point in time <math>t</math> the object was located at <math> \gamma(t) </math> and also how often the object "visited" the point <math> \gamma(t)</math>, * while <math> \mathcal{Crv}(\gamma)</math> is just possible to say, that the object visited the point <math>\gamma(t)</math> on its path <math>\gamma</math> in the topological space <math>\mathbb{R}^n </math>. ==Difference - Graph and Curve== For a path <math>\gamma:[a,b] \to \mathbb{R}^2</math>, the trace or curve is a subset of <math>\mathbb{R}^2</math>, while the graph of the function <math> Graph(\gamma) \subset \mathbb{R}^3 </math>. ==Task - Plot Graph and Curve== Plot the graph [[CAS4Wiki]] and curve of: <math>\begin{array}{rrcl} \gamma: & [0,6\pi] & \rightarrow & \mathbb{R}^2 \ & t & \mapsto & \gamma(t) = \left( 3 \cdot \cos(t), \sin(t) \right) \end{array} </math> ==Animation of the Trace== The following animation shows the animation of a rolling curve of two circles. [[File:Ani Hypocyloid-deltoid.gif|250px|Animation: Rolling Curve]] == Curves in Geogebra == First, create a slider for the variable <math>t \in [0, 2\pi]</math> and two points <math>K_1=(2 \cos(t), 2 \sin(t)) \in \mathbb{R}^2</math> and <math>K_2=(\cos(3 t), \sin(3 t)) \in \mathbb{R}^2</math> and create the sum of the two position vectors <math>K := K_1 + K_2</math>. Analyze the parametrization of the curves. Geogebra: <code> K_1:(2 \cos(t), 2 \sin(t)) </code>, <code> K_2:(\cos(3 t), \sin(3 t)) </code>, <code> K: K_1 + K_2 </code> :<math>\gamma_4(t):= (2 \cos(t), 2 \sin(t)) + (\cos(3 t), \sin(3 t)) \in \mathbb{R}^2 </math> See also [https://www.geogebra.org/m/ppuvs3ge interactive example in Geogebra] == Equation Representations == A curve can also be described by one or more equations in coordinates. The solution set of the equations represents the curve: The equation <math>x^2 + y^2 = 1</math> describes the unit circle in the plane. The equation <math>y^2 = x^2(x + 1)</math> describes the curve given above in parametric form with a double point. If the equation is given by a [[w:en:Polynomial|polynomial]], the curve is called ''[[w:en:Algebraic Curve|algebraic]]''. == Function Graphs == [[w:en:Function graph|Function graphs]] are a special case of both forms mentioned above: The graph of a function <math> f \colon D \to \mathbb{R}, \quad x \mapsto f(x) </math> can either be represented as a parametric form <math> \gamma \colon D \to \mathbb{R}^2, \quad t \mapsto (t, f(t)) </math> or as an equation <math> y = f(x) </math>, where the solution set of the equation represents the curve through <math>{(x, y) \in \mathbb{R}^2 \mid y = f(x)}</math>. When discussing [[w:en:Mathematical analysis|curve analysis]] in [[w:en:Mathematics|school mathematics]], it usually refers to this special case. == Closed Curves == Closed curves <math>\gamma \colon [a,b] \to \mathbb{C}</math> are continuous mappings with <math>\gamma(a) = \gamma(b)</math>. In complex function theory, we need curves <math>\gamma \colon [a,b] \to \mathbb{C}</math> that are continuously differentiable. These are called integration paths. ==Winding Number in Complex Numbers== Smooth closed curves can be assigned an additional number, the [[w:en:Winding number|winding number]], which for a curve parameterized by arc length <math>\gamma \colon [a,b] \to \mathbb{C}</math> is given by: <math>\mu(\gamma, z) := \frac{1}{2\pi i} \int_\gamma \frac{1}{\xi - z} , d\xi := \int_a^b \frac{1}{\gamma(t) - z} \cdot \gamma'(t) , dt</math> The [[w:en:Cauchy’s residue theorem|Cauchy’s residue theorem]] states, analogous to a curve in <math>\mathbb{R}^2</math>, that a simple closed curve has a winding number of <math>1</math> or <math>-1</math>. == Curves as Independent Objects == Curves without an enclosing space are of relatively little interest in [[w:en:Differential geometry|differential geometry]], because every one-dimensional [[w:en:Manifold|manifold]] is [[w:en:Diffeomorphism|diffeomorphic]] to the real line <math>\mathbb{R}</math> or the unit circle <math>S^1</math>. Properties such as the [[w:en:Curvature|curvature]] of a curve are intrinsically not determinable. In [[w:en:Algebraic geometry|algebraic geometry]] and, related to this, in [[w:en:Complex analysis|complex analysis]], "curves" typically refer to one-dimensional [[w:en:Complex manifold|complex manifolds]], often referred to as [[w:en:Riemann surface|Riemann surfaces]]. These curves are independent objects of study, with the most prominent example being [[w:en:Elliptic curve|elliptic curves]]. ''See'' [[w:en:Curve (algebraic geometry)|Curve (algebraic geometry)]]. == Historical Context == The first book of [[w:en:Elements (Euclid)|Euclid's Elements]] begins with the definition: :: "A point is that which has no parts. A curve is a length without width." This definition can no longer be maintained, because there are examples like [[w:en:Peano curve|Peano curves]], i.e., continuous [[w:en:Surjection|surjective]] mappings <math>f \colon \mathbb{R} \to \mathbb{R}^2</math> that fill the entire plane <math>\mathbb{R}^2</math>. On the other hand, according to the [[w:en:Sard’s theorem|Sard’s lemma]], every differentiable curve has zero area, meaning it truly has ''no width,'' as Euclid required. == Interactive Representations of Curves in Geogebra == [https://www.geogebra.org/m/e3hhdrvq Tangent vector of a curve] in <math>\mathbb{R}^2</math> for a curve <math>\gamma:[a,b] \to \mathbb{R}^2</math> with tangent vector <math>\gamma':[a,b] \to \mathbb{R}^2</math> [https://www.geogebra.org/m/srmgcsZX Rolling curves of bicycle reflectors] as an example for curves - [[w:en:Cycloid|Cycloid]] [https://www.geogebra.org/m/ppuvs3ge Rolling curve for example 2] == See also == [[w:en:Space curves in 3D|Space curves in <math>\mathbb{R}^3</math>]] [[Wikipedia:en:Category:Curves in geometry|Curves in Geometry]] [[w:en:Curves in Mathematical analysis|Curves in mathematical analysis]] [[w:en:Cycloid|Cycloid]] == Literature == Ethan D. Bloch: ''A First Course in Geometric Topology and Differential Geometry''. Birkhäuser, Boston 1997. Wilhelm Klingenberg: ''A Course in Differential Geometry''. Springer, New York 1978. == References == <references /> == External Links == {{Commonscat|Curves|Curves}} {{Wiktionary|Curve}} == Page Information == This learning resource can be represented as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slideshow]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Kurs:Funktionentheorie/Kurven&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Kurven&coursetitle=Kurs:Funktionentheorie Wiki2Reveal slideshow]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie Course:Function Theory]'''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was created using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator]. cnvvmqxcn2nwrvkg6nlfu74g53e4p3g 0 317067 2690978 2690845 2024-12-09T06:36:04Z Bert Niehaus 2387134 2690978 wikitext text/x-wiki == Definition: Path == Let <math>U \subset \mathbb{C}</math> be a subset. A path in <math>U</math> is a continuous mapping with: :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. == Definition: Trace of a Path == The trace of a path <math>\gamma \colon [a,b] \rightarrow U </math> in <math>U \subset \mathbb{C}</math> is the image of the function <math>\gamma</math>: :<math>\mathrm{{Spur}}(\gamma):= { \gamma(t) \in \mathbb{C} \ | \ t \in [a,b] }</math> == Definition: Closed Path == Let <math>\gamma \colon [a,b] \rightarrow U </math> be a path in <math>U \subset \mathbb{C}</math>. The mapping <math>\gamma</math> is called a closed path if: :<math>\gamma(a) = \gamma(b)</math> == Definition: Region == Let <math>U \subset \mathbb{C}</math> be an open subset of <math>\mathbb{C}</math>. Then <math>U</math> is called a region. == Definition: Path-Connected == Let <math>U \subset \mathbb{C}</math> be a non-empty set. :<math>U</math> is path-connected <math>:\Longleftrightarrow \ \forall_{z_1,z_2 \in U }\exists_{\gamma\colon [a,b]\rightarrow U}: \ \gamma(a)=z_1 \wedge \gamma(b)=z_2 \wedge{{Spur}}(\gamma) \subseteq U</math> == Definition: Domain == Let <math>G \subset \mathbb{C}</math> be a non-empty subset of <math>\mathbb{C}</math>. If *<math>G</math> is open *<math>G</math> is path-connected Then <math>G</math> is called a domain in <math>\mathbb{C}</math>. == Example (Circular Paths) == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>r > 0</math> be a radius. A circular path <math>\gamma_{z_o,r}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,r}(t):= z_o + r\cdot e^{i\cdot t}</math> == Example - Paths with Ellipse as Trace == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>a, b > 0</math> be the semi-axes of an ellipse. An elliptical path <math>\gamma_{z_o,a,b}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,a,b}(t):= z_o + a\cdot \cos(t) + i\cdot b\cdot \sin(t)</math> == Gardener's Construction of an Ellipse == [[File:Elliko-g.svg|350px|Gardener's Construction of an Ellipse]] == Convex Combinations == Let <math>z_1,z_2 \in \mathbb{C}</math> be complex numbers, and let <math>t \in [0,1]</math> be a scalar. A path <math>\gamma_{z_1,z_2}\colon [0,1] \rightarrow \mathbb{C}</math> is defined such that its trace is the line segment connecting <math>z_1,z_2 \in \mathbb{C}</math>: :<math>\gamma_{z_1,z2}(t):= (1-t)\cdot z_1 + t\cdot z_2</math> Such a path is called a convex combination of the first order (see also [[Convex Combination|Convex Combinations of higher order]]). === Animation of a Convex Combination of Two Vectors as Mapping === [[File:Convex combination 1 ord with geogebra.gif|450px|center|Convex Combination as Mapping in an Animated GIF]] == Integration Path == Let <math>G \subset \mathbb{C}</math> be a domain. An integration path in <math>G</math> is a path that is piecewise continuously differentiable with :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. === Remark === An integration path can, for example, be expressed piecewise as convex combinations between multiple points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The overall path does not need to be differentiable at points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The trace of such a path is also called a polygonal path. == See Also == * [[w:en:Ellipse|Ellipse]] * [[Convex Combination]] * [[Paths in Topological Vector Spaces]] 6a9vzwfpvosexygookjz35ef8y8t1of 2690981 2690978 2024-12-09T07:16:54Z Bert Niehaus 2387134 2690981 wikitext text/x-wiki == Definition: Path == Let <math>U \subset \mathbb{C}</math> be a subset. A path in <math>U</math> is a [[w:en:continuous function|continuous mapping]] with: :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. == Definition: Trace of a Path == The trace of a path <math>\gamma \colon [a,b] \rightarrow U </math> in <math>U \subset \mathbb{C}</math> is the [[w:en:Range of a function|image or range]] of the function <math>\gamma</math>: :<math>\mathrm{{Trace}}(\gamma):= { \gamma(t) \in \mathbb{C} \ | \ t \in [a,b] }</math> == Definition: Closed Path == Let <math>\gamma \colon [a,b] \rightarrow U </math> be a path in <math>U \subset \mathbb{C}</math>. The mapping <math>\gamma</math> is called a closed path if: :<math>\gamma(a) = \gamma(b)</math> == Definition: Region == Let <math>U \subset \mathbb{C}</math> be an open subset of <math>\mathbb{C}</math>. Then <math>U</math> is called a region. == Definition: Path-Connected == Let <math>U \subset \mathbb{C}</math> be a non-empty set. :<math>U</math> is path-connected <math>:\Longleftrightarrow \ \forall_{z_1,z_2 \in U }\exists_{\gamma\colon [a,b]\rightarrow U}: \ \gamma(a)=z_1 \wedge \gamma(b)=z_2 \wedge{{Spur}}(\gamma) \subseteq U</math> == Definition: Domain == Let <math>G \subset \mathbb{C}</math> be a non-empty subset of <math>\mathbb{C}</math>. If *<math>G</math> is open *<math>G</math> is path-connected Then <math>G</math> is called a domain in <math>\mathbb{C}</math>. == Example (Circular Paths) == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>r > 0</math> be a radius. A circular path <math>\gamma_{z_o,r}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,r}(t):= z_o + r\cdot e^{i\cdot t}</math> == Example - Paths with Ellipse as Trace == Let <math>z_o \in \mathbb{C}</math> be a complex number, and let <math>a, b > 0</math> be the semi-axes of an ellipse. An elliptical path <math>\gamma_{z_o,a,b}\colon [0,2\pi] \rightarrow \mathbb{C}</math> around <math>z_o \in \mathbb{C}</math> is defined as: :<math>\gamma_{z_o,a,b}(t):= z_o + a\cdot \cos(t) + i\cdot b\cdot \sin(t)</math> == Gardener's Construction of an Ellipse == [[File:Elliko-g.svg|350px|Gardener's Construction of an Ellipse]] == Convex Combinations == Let <math>z_1,z_2 \in \mathbb{C}</math> be complex numbers, and let <math>t \in [0,1]</math> be a scalar. A path <math>\gamma_{z_1,z_2}\colon [0,1] \rightarrow \mathbb{C}</math> is defined such that its trace is the line segment connecting <math>z_1,z_2 \in \mathbb{C}</math>: :<math>\gamma_{z_1,z2}(t):= (1-t)\cdot z_1 + t\cdot z_2</math> Such a path is called a convex combination of the first order (see also [[Convex Combination|Convex Combinations of higher order]]). === Animation of a Convex Combination of Two Vectors as Mapping === [[File:Convex combination 1 ord with geogebra.gif|450px|center|Convex Combination as Mapping in an Animated GIF]] == Integration Path == Let <math>G \subset \mathbb{C}</math> be a domain. An integration path in <math>G</math> is a path that is piecewise continuously differentiable with :<math>\gamma \colon [a,b] \rightarrow U </math> with <math>a < b</math> and <math>a,b \in \mathbb{R}</math>. === Remark === An integration path can, for example, be expressed piecewise as convex combinations between multiple points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The overall path does not need to be differentiable at points <math>z_1, \ldots z_n \in \mathbb{C}</math>. The trace of such a path is also called a polygonal path. == See Also == * [[w:en:Ellipse|Ellipse]] * [[Convex Combination]] * [[Paths in Topological Vector Spaces]] blxgbt3jeflbkhby8dz5xus7vp9wthl 0 317068 2690891 2690753 2024-12-08T18:45:22Z Eshaa2024 2993595 2690891 wikitext text/x-wiki == Introduction == This page on the topic ''Path Integral'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slides]'''. Individual sections are considered as slides, and changes to the slides immediately affect the content. The following aspects are discussed in detail: *(1) Paths as continuous mappings from an interval <math>[a,b]</math> into the complex numbers <math>\mathbb{C}</math> over which integration occurs, *(2) Derivatives of curves/paths as prerequisites for the definition of path integrals, *(3) Definition of path integrals. == learnig requiremets == The learning resource on the topic ''Path Integral'' has the following prerequisites, which are helpful or necessary for understanding the following explanation: *Concept of [[w:en:Course:Measure_theory_on_topological_spaces/Paths_in_topological_spaces|path in a topological space]], *Differentiability in real analysis, *Integration in real analysis. == Basic Geometric Idea of the Path Integral == The following curve <math>\gamma</math> loops around a point <math>z_0\in \mathbb{C}</math> twice. [[File:Windungszahl5.png|150px|center|Path around a point]] == Integral over an Interval == Let <math>G\subseteq \mathbb{C}</math> be a [[w:en:Domain (mathematics)|domain]] and <math>g\colon [a,b] \to\mathbb{C}</math> a [[w:en:Complex function|complex-valued function]]. The function <math>g</math> is called integrable if ::<math>\operatorname{Re}(g):G \to\mathbb{R}</math> and <math>\operatorname{Im}(g):G \to\mathbb{R}</math> with <math>g=\operatorname{Re}(g) + i \cdot \operatorname{Im}(g)</math> are integrable functions. It is defined as :<math>\int\limits_a^b g(x)\mathrm{d} x := \int\limits_a^b\operatorname{Re}(g)(x)\mathrm{d}x +\mathrm{i}\int\limits_a^b\operatorname{Im}(g)(x)\mathrm{d}x</math>. Thus, the integral is <math>\mathbb{C}</math>-linear. If <math>g</math> is continuous and <math>G</math> is an antiderivative of <math>g</math>, then as in the real case, :<math>\int\limits_a^b g(x)\mathrm{d}x = G(b)-G(a)</math>. == Extension of the Integral Concept == The integral concept is extended through the definition of an integration path in the complex plane as follows: If <math>f\colon G\to\mathbb{C}</math> is a complex-valued function on a [[w:en:Domain (mathematics)|domain]] <math>G\subseteq\mathbb{C}</math>, and <math>\gamma\colon[a,b]\to G</math> is a piecewise continuously differentiable [[w:en:Path (mathematics)|path]] in <math>G</math>, then the ''path integral'' of <math>f</math> along the path <math>\gamma</math> is defined as : <math>\int\limits_\gamma f:=\int\limits_\gamma f(z),\mathrm dz:=\int\limits_a^b f(\gamma(t))\cdot \gamma'(t),\mathrm dt.</math> Here, the multiplication sign refers to complex multiplication.<ref>„Curve Integral“. In: Wikipedia, The Free Encyclopedia. Editing status: November 24, 2017, 16:22 UTC. URL: https://en.wikipedia.org/w/index.php?title=Curve_integral&oldid=171345033 (Accessed: December 8, 2017, 14:27 UTC) </ref> == Cauchy's Integral Theorem == The central statement about path integrals of complex functions is the [[w:en:Cauchy's_integral_theorem|Cauchy Integral Theorem]]: For a [[w:en:Holomorphic function|holomorphic]] function <math>f</math>, the path integral depends only on the [[w:en:Homotopy|homotopy]] class of <math>\gamma</math>. If <math>U</math> is [[w:en:Simply_connected_space|simply connected]], then the integral depends not on <math>\gamma</math>, but only on the starting and ending points. Analogous to the real case, the ''length'' of the path <math>\gamma:[a,b]\rightarrow \mathbb{C}</math> is defined as :<math>\operatorname{L}(\gamma):=\int\limits_a^b \left| \gamma'(t) \right| \mathrm{d}t</math>. For theoretical purposes, the following inequality, called the ''standard estimate'', is of particular interest: :<math>\left| \int_\gamma f(z) , \mathrm dz \right| \leq \operatorname{L}(\gamma)\cdot C</math>, if <math>\left| f(z) \right|\leq C</math> for all <math>z\in\gamma([0,1])</math>. As in the real case, the path integral is independent of the parametrization of the path <math>\gamma</math>, i.e., it is not strictly necessary to choose <math>[0,1]</math> as the parameter domain, as can be shown by substitution. This allows the definition of complex curve integrals by replacing the above formulas with a curve <math>\mathcal{C}</math> in <math>\mathbb{C}</math>. == Exercises == *Be <math>\gamma\colon[a,b]\to G</math> with <math>t\mapsto \gamma(t)= \sin(t)+i\cdot t^2</math>. Determine <math>\gamma'(t)</math>! *Compute the path integral <math>\int\limits_\gamma \frac{1} {z},\mathrm dz</math> for the path <math>\gamma\colon[0,2\pi] \to \mathbb{C}</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. *Calculate the length of the path <math>L(\gamma)</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. == See also == [[w:en:Kurs:Funktionentheorie]] [[w:en:Curve integral]] == References == <references/> == Page Information == This learning resource can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the learning unit '''[https://en.wikiversity.org/wiki/Course:Function_theory Course:Function_theory]''' the link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was created using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. [https://en.wikiversity.org/wiki/Wegintegral The page] was created as document type [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source in Wikiversity: https://en.wikiversity.org/wiki/Wegintegral See also further information on [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal link generator]. 9gxme0kl0fzigavcqrrslft2856v12g 2690897 2690891 2024-12-08T20:33:21Z Eshaa2024 2993595 2690897 wikitext text/x-wiki == Introduction == This page on the topic "Path Integral" can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slides]'''. Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides. The following subtopics are treated in detail: (1) Paths as continuous mappings from an interval <math>[a,b]</math> into the complex numbers <math>\mathbb{C}</math> over which integration is performed, (2) Derivatives of curves/paths as a prerequisite for the definition of path integrals, (3) Definition of path integrals == Learning Prerequisites == The learning resource on the topic "Path Integral" has the following learning prerequisites, which are helpful or necessary for understanding the subsequent explanations: *Concept of [[w:en Path in a topological space]], *Differentiability in real analysis, *Integration in real analysis. == Basic Geometric Idea of the Path Integral == The following curve <math>\gamma</math> loops around a point <math>z_0\in \mathbb{C}</math> twice. [[File:Windungszahl5.png|150px|center|Path around a point]] == Integral over an Interval == Let <math>G\subseteq \mathbb{C}</math> be a [[w:en:Domain (mathematics)|domain]] and <math>g\colon [a,b] \to\mathbb{C}</math> a [[w:en:Complex function|complex-valued function]]. The function <math>g</math> is called integrable if ::<math>\operatorname{Re}(g):G \to\mathbb{R}</math> and <math>\operatorname{Im}(g):G \to\mathbb{R}</math> with <math>g=\operatorname{Re}(g) + i \cdot \operatorname{Im}(g)</math> are integrable functions. It is defined as :<math>\int\limits_a^b g(x)\mathrm{d} x := \int\limits_a^b\operatorname{Re}(g)(x)\mathrm{d}x +\mathrm{i}\int\limits_a^b\operatorname{Im}(g)(x)\mathrm{d}x</math>. Thus, the integral is <math>\mathbb{C}</math>-linear. If <math>g</math> is continuous and <math>G</math> is an antiderivative of <math>g</math>, then as in the real case, :<math>\int\limits_a^b g(x)\mathrm{d}x = G(b)-G(a)</math>. == Extension of the Integral Concept == The integral concept is extended through the definition of an integration path in the complex plane as follows: If <math>f\colon G\to\mathbb{C}</math> is a complex-valued function on a [[w:en:Domain (mathematics)|domain]] <math>G\subseteq\mathbb{C}</math>, and <math>\gamma\colon[a,b]\to G</math> is a piecewise continuously differentiable [[w:en:Path (mathematics)|path]] in <math>G</math>, then the ''path integral'' of <math>f</math> along the path <math>\gamma</math> is defined as : <math>\int\limits_\gamma f:=\int\limits_\gamma f(z),\mathrm dz:=\int\limits_a^b f(\gamma(t))\cdot \gamma'(t),\mathrm dt.</math> Here, the multiplication sign refers to complex multiplication.<ref>„Curve Integral“. In: Wikipedia, The Free Encyclopedia. Editing status: November 24, 2017, 16:22 UTC. URL: https://en.wikipedia.org/w/index.php?title=Curve_integral&oldid=171345033 (Accessed: December 8, 2017, 14:27 UTC) </ref> == Cauchy's Integral Theorem == The central statement about path integrals of complex functions is the [[w:en:Cauchy's_integral_theorem|Cauchy Integral Theorem]]: For a [[w:en:Holomorphic function|holomorphic]] function <math>f</math>, the path integral depends only on the [[w:en:Homotopy|homotopy]] class of <math>\gamma</math>. If <math>U</math> is [[w:en:Simply_connected_space|simply connected]], then the integral depends not on <math>\gamma</math>, but only on the starting and ending points. Analogous to the real case, the ''length'' of the path <math>\gamma:[a,b]\rightarrow \mathbb{C}</math> is defined as :<math>\operatorname{L}(\gamma):=\int\limits_a^b \left| \gamma'(t) \right| \mathrm{d}t</math>. For theoretical purposes, the following inequality, called the ''standard estimate'', is of particular interest: :<math>\left| \int_\gamma f(z) , \mathrm dz \right| \leq \operatorname{L}(\gamma)\cdot C</math>, if <math>\left| f(z) \right|\leq C</math> for all <math>z\in\gamma([0,1])</math>. As in the real case, the path integral is independent of the parametrization of the path <math>\gamma</math>, i.e., it is not strictly necessary to choose <math>[0,1]</math> as the parameter domain, as can be shown by substitution. This allows the definition of complex curve integrals by replacing the above formulas with a curve <math>\mathcal{C}</math> in <math>\mathbb{C}</math>. == Exercises == *Be <math>\gamma\colon[a,b]\to G</math> with <math>t\mapsto \gamma(t)= \sin(t)+i\cdot t^2</math>. Determine <math>\gamma'(t)</math>! *Compute the path integral <math>\int\limits_\gamma \frac{1} {z},\mathrm dz</math> for the path <math>\gamma\colon[0,2\pi] \to \mathbb{C}</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. *Calculate the length of the path <math>L(\gamma)</math> with <math>t\mapsto \gamma(t)= r\cdot e^{i\cdot t}</math>. == See also == [[w:en Function theory (course)|Function Theory Course]] [[w:en Contour integral|Contour Integral]] == References == <references/> == Page Information == This learning resource can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=de&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''. === Wiki2Reveal === This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the learning unit '''[https://en.wikiversity.org/wiki/Course:Function_theory Course:Function_theory]''' the link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was created using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator]. [https://en.wikiversity.org/wiki/Wegintegral The page] was created as document type [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE]. Link to the source in Wikiversity: https://en.wikiversity.org/wiki/Wegintegral See also further information on [[v:en:Wiki2Reveal|Wiki2Reveal]] and at [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&title=Wegintegral&author=Kurs:Funktionentheorie&language=en&audioslide=yes&shorttitle=Wegintegral&coursetitle=Kurs:Funktionentheorie Wiki2Reveal link generator]. 2qgs983gm4ij3sqczde1685eq1aocqy 3 317079 2690876 2024-12-08T15:58:01Z Ottawahitech 2369270 /* Several pages in Hobo tourism and Hobo travel journalism-deletion at wb thepges */ new section 2690876 wikitext text/x-wiki == Several pages in Hobo tourism and Hobo travel journalism-deletion at wb thepges == Hi {{PAGENAME}}, I saw your [[posting at the Colloquium]], and that led me to a discussion about deleting these pages mentioned in the title at wikibooks. I wonder if Wikivoyage may be interested? cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:58, 8 December 2024 (UTC) glvdfcn6xj9lv3lr5uw1czczmz6ltxd 2690877 2690876 2024-12-08T16:00:52Z Ottawahitech 2369270 /* Several pages in Hobo tourism and Hobo travel journalism-deletion at wb thepges */ 2690877 wikitext text/x-wiki == Several pages in Hobo tourism and Hobo travel journalism-deletion at wb thepges == Hi {{PAGENAME}}, I saw your [[Wikiversity:Colloquium#Import Resource From Wikibooks?|posting at the Colloquium]], and that led me to a discussion about deleting these pages mentioned in the title at wikibooks. I wonder if Wikivoyage may be interested? cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:58, 8 December 2024 (UTC) 5thghstvaq213g2rnqob2w3n132rtng 3 317080 2690886 2024-12-08T18:17:11Z Atcovi 276019 /* Welcome */ new section 2690886 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], TiwariA.23!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:17, 8 December 2024 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} o9a8kfz4vm9pnfqtcf2dwt98ka5g879 2690894 2690886 2024-12-08T19:29:15Z Atcovi 276019 /* Subject Summary for Aaryan Tiwari (8CJB) */ new section 2690894 wikitext text/x-wiki ==Welcome== {{Robelbox|theme=9|title='''[[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]], TiwariA.23!'''|width=100%}} <div style="{{Robelbox/pad}}"> You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or get in touch with [[User talk:Atcovi|me personally]] if you would like some [[Help:Contents|help]]. Remember to [[Wikiversity:Signature#How to add your signature|sign]] your comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. Using the signature icon [[File:OOjs UI icon signature-ltr.svg]] makes it simple. We invite you to [[Wikiversity:Be bold|be bold]] and [[Wikiversity|assume good faith]]. Please abide by our [[Wikiversity:Civility|civility]], [[Wikiversity:Privacy policy|privacy]], and [[Foundation:Terms of Use|terms of use]] policies. To find your way around, check out: <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Wikiversity:Introduction|Introduction to Wikiversity]] * [[Help:Guides|Take a guided tour]] and learn [[Help:Editing|how to edit]] * [[Wikiversity:Browse|Browse]] or visit an educational level portal:<br>[[Portal:Pre-school Education|pre-school]] | [[Portal:Primary Education|primary]] | [[Portal:Secondary Education|secondary]] | [[Portal:Tertiary Education|tertiary]] | [[Portal:Non-formal Education|non-formal]] * [[Wikiversity:Introduction explore|Explore]] links in left-hand navigation menu </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] * Learn [[Help:How to write an educational resource|how to write an educational resource]] * Find out about [[Wikiversity:Research|research]] activities * Give [[Wikiversity:Feedback|feedback]] about your observations * Discuss issues or ask questions at the [[Wikiversity:Colloquium|colloquium]] </div> <br clear="both"/> To get started, experiment in the [[wikiversity:sandbox|sandbox]] or on [[special:mypage|your userpage]]. See you around Wikiversity! --—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:17, 8 December 2024 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} == [[Subject Summary for Aaryan Tiwari (8CJB)]] == Hello. I'm not sure why you decided to revert my edit where I moved this page into your userspace, but I've gone ahead and deleted the page as it seemed to be a test page and had no [[WV:What is Wikiversity?|learning value]]. Could you please clarify your revert and what you intend to make out of this page? 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indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="DT"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below" |- !colspan="4"| <span class="senior">3808</span> <span class="junior">14</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="DT"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |} ===size 12=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below" |- !colspan="4"| <span class="senior">680</span> <span class="junior">2</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="DT"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below" |- !colspan="4"| <span class="senior">6856</span> <span class="junior">26</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices 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ngfwu0514quaqrx34daiwf170gfih91 2690918 2690911 2024-12-08T22:02:22Z Watchduck 137431 2690918 wikitext text/x-wiki <templatestyles src="Collapsible with classes/style.css" /> <templatestyles src="Principalities and dominions/overview/style.css" /> ===size 1=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">0</span> <span class="junior">0</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 1; king index 0.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 1; king index 0.svg|200px]] |class="DT"| [[File:3T dominion; faction size 1; king index 0.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 1; king index 0.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below 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size 3; king index 6.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="DT"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">28384</span> <span class="junior">110</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="DT"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |} ===size 4=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">5760</span> <span class="junior">22</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="DT"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">10920</span> <span class="junior">42</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="DT"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |} ===size 6=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">2176</span> <span class="junior">8</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="DT"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">3808</span> <span class="junior">14</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="DT"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |} ===size 12=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">680</span> <span class="junior">2</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="DT"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">6856</span> <span class="junior">26</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="DT"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="senior">11464</span> <span class="junior">44</span> |- class="light-heading bold" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="PT"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="PZ"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="DT"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="DZ"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |} gin1ytsgkpexrdpximotrnqz6fexehb 2690920 2690918 2024-12-08T22:53:03Z Watchduck 137431 2690920 wikitext text/x-wiki <templatestyles src="Collapsible with classes/style.css" /> <templatestyles src="Principalities and dominions/overview/style.css" /> ===size 1=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 0</span> <span class="senior">(king 0)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 0.svg|200px]] |- |<span class="venn">[[File:Venn 0000 0000.svg|20px]]<br><span class="light">(</span>0<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0000.svg|20px]]<br><span class="light">(</span>0<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0000.svg|20px]]<br><span class="light">(</span>0<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0000.svg|20px]]<br><span class="light">(</span>0<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 104</span> <span class="senior">(king 26752)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 104.svg|200px]] |- |<span class="venn">[[File:Venn 0001 0110.svg|20px]]<br><span class="light">(</span>104<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 0111.svg|20px]]<br><span class="light">(</span>232<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1001.svg|20px]]<br><span class="light">(</span>150<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1000.svg|20px]]<br><span class="light">(</span>22<span class="light">)</span></span> |} ===size 3=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 6</span> <span class="senior">(king 1632)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |- |<span class="venn">[[File:Venn 0110 0000.svg|20px]]<br><span class="light">(</span>6<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1000.svg|20px]]<br><span class="light">(</span>18<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1000.svg|20px]]<br><span class="light">(</span>20<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 0110.svg|20px]]<br><span class="light">(</span>102<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1010.svg|20px]]<br><span class="light">(</span>90<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1100.svg|20px]]<br><span class="light">(</span>60<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0110.svg|20px]]<br><span class="light">(</span>96<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0010.svg|20px]]<br><span class="light">(</span>72<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0100.svg|20px]]<br><span class="light">(</span>40<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0110.svg|20px]]<br><span class="light">(</span>96<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0010.svg|20px]]<br><span class="light">(</span>72<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0100.svg|20px]]<br><span class="light">(</span>40<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 110</span> <span class="senior">(king 28384)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |- |<span class="venn">[[File:Venn 0111 0110.svg|20px]]<br><span class="light">(</span>110<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1110.svg|20px]]<br><span class="light">(</span>122<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1110.svg|20px]]<br><span class="light">(</span>124<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 0001.svg|20px]]<br><span class="light">(</span>142<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1101.svg|20px]]<br><span class="light">(</span>178<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1011.svg|20px]]<br><span class="light">(</span>212<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1111.svg|20px]]<br><span class="light">(</span>246<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1011.svg|20px]]<br><span class="light">(</span>222<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1101.svg|20px]]<br><span class="light">(</span>190<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1110.svg|20px]]<br><span class="light">(</span>118<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1010.svg|20px]]<br><span class="light">(</span>94<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1100.svg|20px]]<br><span class="light">(</span>62<span class="light">)</span></span> |} ===size 4=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 22</span> <span class="senior">(king 5760)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |- |<span class="venn">[[File:Venn 0110 1000.svg|20px]]<br><span class="light">(</span>22<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1001.svg|20px]]<br><span class="light">(</span>150<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 0111.svg|20px]]<br><span class="light">(</span>232<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 0110.svg|20px]]<br><span class="light">(</span>104<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0001 1000.svg|20px]]<br><span class="light">(</span>24<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0100.svg|20px]]<br><span class="light">(</span>36<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0010.svg|20px]]<br><span class="light">(</span>66<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1110.svg|20px]]<br><span class="light">(</span>120<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0110.svg|20px]]<br><span class="light">(</span>108<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0110.svg|20px]]<br><span class="light">(</span>106<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1110.svg|20px]]<br><span class="light">(</span>120<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0110.svg|20px]]<br><span class="light">(</span>108<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0110.svg|20px]]<br><span class="light">(</span>106<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1000.svg|20px]]<br><span class="light">(</span>24<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0100.svg|20px]]<br><span class="light">(</span>36<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0010.svg|20px]]<br><span class="light">(</span>66<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 42</span> <span class="senior">(king 10920)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |- |<span class="venn">[[File:Venn 0111 1110.svg|20px]]<br><span class="light">(</span>126<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 1110.svg|20px]]<br><span class="light">(</span>126<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 1110.svg|20px]]<br><span class="light">(</span>126<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 1110.svg|20px]]<br><span class="light">(</span>126<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0101 0100.svg|20px]]<br><span class="light">(</span>42<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0010.svg|20px]]<br><span class="light">(</span>76<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1110.svg|20px]]<br><span class="light">(</span>112<span class="light">)</span></span> |<span class="venn">[[File:Venn 0100 0001.svg|20px]]<br><span class="light">(</span>130<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0001.svg|20px]]<br><span class="light">(</span>132<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1001.svg|20px]]<br><span class="light">(</span>144<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1111.svg|20px]]<br><span class="light">(</span>252<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1111.svg|20px]]<br><span class="light">(</span>250<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0111.svg|20px]]<br><span class="light">(</span>238<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1010.svg|20px]]<br><span class="light">(</span>84<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1100.svg|20px]]<br><span class="light">(</span>50<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0000.svg|20px]]<br><span class="light">(</span>14<span class="light">)</span></span> |} ===size 6=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 8</span> <span class="senior">(king 2176)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |- |<span class="venn">[[File:Venn 0001 0000.svg|20px]]<br><span class="light">(</span>8<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0100.svg|20px]]<br><span class="light">(</span>32<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0010.svg|20px]]<br><span class="light">(</span>64<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 0001.svg|20px]]<br><span class="light">(</span>136<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0101.svg|20px]]<br><span class="light">(</span>160<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0011.svg|20px]]<br><span class="light">(</span>192<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1111.svg|20px]]<br><span class="light">(</span>240<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0011.svg|20px]]<br><span class="light">(</span>204<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0101.svg|20px]]<br><span class="light">(</span>170<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1000.svg|20px]]<br><span class="light">(</span>16<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0000.svg|20px]]<br><span class="light">(</span>4<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0000.svg|20px]]<br><span class="light">(</span>2<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0001 0100.svg|20px]]<br><span class="light">(</span>40<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0010.svg|20px]]<br><span class="light">(</span>72<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0110.svg|20px]]<br><span class="light">(</span>96<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 0100.svg|20px]]<br><span class="light">(</span>40<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0010.svg|20px]]<br><span class="light">(</span>72<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0110.svg|20px]]<br><span class="light">(</span>96<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1100.svg|20px]]<br><span class="light">(</span>60<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1010.svg|20px]]<br><span class="light">(</span>90<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0110.svg|20px]]<br><span class="light">(</span>102<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1000.svg|20px]]<br><span class="light">(</span>20<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1000.svg|20px]]<br><span class="light">(</span>18<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0000.svg|20px]]<br><span class="light">(</span>6<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 14</span> <span class="senior">(king 3808)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |- |<span class="venn">[[File:Venn 0111 0000.svg|20px]]<br><span class="light">(</span>14<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1100.svg|20px]]<br><span class="light">(</span>50<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1010.svg|20px]]<br><span class="light">(</span>84<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 0111.svg|20px]]<br><span class="light">(</span>238<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1111.svg|20px]]<br><span class="light">(</span>250<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1111.svg|20px]]<br><span class="light">(</span>252<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1001.svg|20px]]<br><span class="light">(</span>144<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0001.svg|20px]]<br><span class="light">(</span>132<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0001.svg|20px]]<br><span class="light">(</span>130<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1110.svg|20px]]<br><span class="light">(</span>112<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0010.svg|20px]]<br><span class="light">(</span>76<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0100.svg|20px]]<br><span class="light">(</span>42<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0011 1100.svg|20px]]<br><span class="light">(</span>60<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1010.svg|20px]]<br><span class="light">(</span>90<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0110.svg|20px]]<br><span class="light">(</span>102<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1000.svg|20px]]<br><span class="light">(</span>20<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1000.svg|20px]]<br><span class="light">(</span>18<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0000.svg|20px]]<br><span class="light">(</span>6<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1000.svg|20px]]<br><span class="light">(</span>20<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1000.svg|20px]]<br><span class="light">(</span>18<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0000.svg|20px]]<br><span class="light">(</span>6<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1100.svg|20px]]<br><span class="light">(</span>60<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1010.svg|20px]]<br><span class="light">(</span>90<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0110.svg|20px]]<br><span class="light">(</span>102<span class="light">)</span></span> |} ===size 12=== {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 2</span> <span class="senior">(king 680)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |- |<span class="venn">[[File:Venn 0100 0000.svg|20px]]<br><span class="light">(</span>2<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0000.svg|20px]]<br><span class="light">(</span>4<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1000.svg|20px]]<br><span class="light">(</span>16<span class="light">)</span></span> |<span class="venn">[[File:Venn 0101 0101.svg|20px]]<br><span class="light">(</span>170<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0011.svg|20px]]<br><span class="light">(</span>204<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1111.svg|20px]]<br><span class="light">(</span>240<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0011.svg|20px]]<br><span class="light">(</span>192<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0101.svg|20px]]<br><span class="light">(</span>160<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0001.svg|20px]]<br><span class="light">(</span>136<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 0010.svg|20px]]<br><span class="light">(</span>64<span class="light">)</span></span><span class="venn">[[File:Venn 0000 0100.svg|20px]]<br><span class="light">(</span>32<span class="light">)</span></span><span class="venn">[[File:Venn 0001 0000.svg|20px]]<br><span class="light">(</span>8<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0111 1000.svg|20px]]<br><span class="light">(</span>30<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1100.svg|20px]]<br><span class="light">(</span>54<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1010.svg|20px]]<br><span class="light">(</span>86<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 1000.svg|20px]]<br><span class="light">(</span>30<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1100.svg|20px]]<br><span class="light">(</span>54<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1010.svg|20px]]<br><span class="light">(</span>86<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1000.svg|20px]]<br><span class="light">(</span>24<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0100.svg|20px]]<br><span class="light">(</span>36<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0010.svg|20px]]<br><span class="light">(</span>66<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1110.svg|20px]]<br><span class="light">(</span>120<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0110.svg|20px]]<br><span class="light">(</span>108<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0110.svg|20px]]<br><span class="light">(</span>106<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0101 0000.svg|20px]]<br><span class="light">(</span>10<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0000.svg|20px]]<br><span class="light">(</span>12<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0100.svg|20px]]<br><span class="light">(</span>34<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1100.svg|20px]]<br><span class="light">(</span>48<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0010.svg|20px]]<br><span class="light">(</span>68<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1010.svg|20px]]<br><span class="light">(</span>80<span class="light">)</span></span> |<span class="venn">[[File:Venn 0100 0100.svg|20px]]<br><span class="light">(</span>34<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0010.svg|20px]]<br><span class="light">(</span>68<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0000.svg|20px]]<br><span class="light">(</span>10<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1010.svg|20px]]<br><span class="light">(</span>80<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0000.svg|20px]]<br><span class="light">(</span>12<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1100.svg|20px]]<br><span class="light">(</span>48<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1100.svg|20px]]<br><span class="light">(</span>48<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1010.svg|20px]]<br><span class="light">(</span>80<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0000.svg|20px]]<br><span class="light">(</span>12<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0010.svg|20px]]<br><span class="light">(</span>68<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0000.svg|20px]]<br><span class="light">(</span>10<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0100.svg|20px]]<br><span class="light">(</span>34<span class="light">)</span></span> |<span class="venn">[[File:Venn 0000 1010.svg|20px]]<br><span class="light">(</span>80<span class="light">)</span></span><span class="venn">[[File:Venn 0000 1100.svg|20px]]<br><span class="light">(</span>48<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0010.svg|20px]]<br><span class="light">(</span>68<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0000.svg|20px]]<br><span class="light">(</span>12<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0100.svg|20px]]<br><span class="light">(</span>34<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0000.svg|20px]]<br><span class="light">(</span>10<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 26</span> <span class="senior">(king 6856)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |- |<span class="venn">[[File:Venn 0101 1000.svg|20px]]<br><span class="light">(</span>26<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1000.svg|20px]]<br><span class="light">(</span>28<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0100.svg|20px]]<br><span class="light">(</span>38<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1100.svg|20px]]<br><span class="light">(</span>52<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0010.svg|20px]]<br><span class="light">(</span>70<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1010.svg|20px]]<br><span class="light">(</span>82<span class="light">)</span></span> |<span class="venn">[[File:Venn 0100 1011.svg|20px]]<br><span class="light">(</span>210<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1101.svg|20px]]<br><span class="light">(</span>180<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0011.svg|20px]]<br><span class="light">(</span>198<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1001.svg|20px]]<br><span class="light">(</span>156<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0101.svg|20px]]<br><span class="light">(</span>166<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1001.svg|20px]]<br><span class="light">(</span>154<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1101.svg|20px]]<br><span class="light">(</span>184<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1011.svg|20px]]<br><span class="light">(</span>216<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0101.svg|20px]]<br><span class="light">(</span>172<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0111.svg|20px]]<br><span class="light">(</span>228<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0011.svg|20px]]<br><span class="light">(</span>202<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0111.svg|20px]]<br><span class="light">(</span>226<span class="light">)</span></span> |<span class="venn">[[File:Venn 0001 1010.svg|20px]]<br><span class="light">(</span>88<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1100.svg|20px]]<br><span class="light">(</span>56<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0110.svg|20px]]<br><span class="light">(</span>100<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0100.svg|20px]]<br><span class="light">(</span>44<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0110.svg|20px]]<br><span class="light">(</span>98<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0010.svg|20px]]<br><span class="light">(</span>74<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0111 0100.svg|20px]]<br><span class="light">(</span>46<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1100.svg|20px]]<br><span class="light">(</span>58<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0010.svg|20px]]<br><span class="light">(</span>78<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1010.svg|20px]]<br><span class="light">(</span>92<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1110.svg|20px]]<br><span class="light">(</span>114<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1110.svg|20px]]<br><span class="light">(</span>116<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 0010.svg|20px]]<br><span class="light">(</span>78<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1110.svg|20px]]<br><span class="light">(</span>114<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0100.svg|20px]]<br><span class="light">(</span>46<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1110.svg|20px]]<br><span class="light">(</span>116<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1100.svg|20px]]<br><span class="light">(</span>58<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1010.svg|20px]]<br><span class="light">(</span>92<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1010.svg|20px]]<br><span class="light">(</span>92<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1110.svg|20px]]<br><span class="light">(</span>116<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1100.svg|20px]]<br><span class="light">(</span>58<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1110.svg|20px]]<br><span class="light">(</span>114<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0100.svg|20px]]<br><span class="light">(</span>46<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0010.svg|20px]]<br><span class="light">(</span>78<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1110.svg|20px]]<br><span class="light">(</span>116<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1010.svg|20px]]<br><span class="light">(</span>92<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1110.svg|20px]]<br><span class="light">(</span>114<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1100.svg|20px]]<br><span class="light">(</span>58<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0010.svg|20px]]<br><span class="light">(</span>78<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0100.svg|20px]]<br><span class="light">(</span>46<span class="light">)</span></span> |} {| class="collapsible-with-classes collapsible collapsed wide center gap-below wikitable" |- !colspan="4"| <span class="junior">king index 44</span> <span class="senior">(king 11464)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |- |<span class="venn">[[File:Venn 0111 1100.svg|20px]]<br><span class="light">(</span>62<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1010.svg|20px]]<br><span class="light">(</span>94<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1110.svg|20px]]<br><span class="light">(</span>118<span class="light">)</span></span> |<span class="venn">[[File:Venn 0111 1101.svg|20px]]<br><span class="light">(</span>190<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1011.svg|20px]]<br><span class="light">(</span>222<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1111.svg|20px]]<br><span class="light">(</span>246<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1011.svg|20px]]<br><span class="light">(</span>212<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1101.svg|20px]]<br><span class="light">(</span>178<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0001.svg|20px]]<br><span class="light">(</span>142<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1110.svg|20px]]<br><span class="light">(</span>124<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1110.svg|20px]]<br><span class="light">(</span>122<span class="light">)</span></span><span class="venn">[[File:Venn 0111 0110.svg|20px]]<br><span class="light">(</span>110<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0101 0110.svg|20px]]<br><span class="light">(</span>106<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0110.svg|20px]]<br><span class="light">(</span>108<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1110.svg|20px]]<br><span class="light">(</span>120<span class="light">)</span></span> |<span class="venn">[[File:Venn 0100 0010.svg|20px]]<br><span class="light">(</span>66<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0100.svg|20px]]<br><span class="light">(</span>36<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1000.svg|20px]]<br><span class="light">(</span>24<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1010.svg|20px]]<br><span class="light">(</span>86<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1100.svg|20px]]<br><span class="light">(</span>54<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1000.svg|20px]]<br><span class="light">(</span>30<span class="light">)</span></span> |<span class="venn">[[File:Venn 0110 1010.svg|20px]]<br><span class="light">(</span>86<span class="light">)</span></span><span class="venn">[[File:Venn 0110 1100.svg|20px]]<br><span class="light">(</span>54<span class="light">)</span></span><span class="venn">[[File:Venn 0111 1000.svg|20px]]<br><span class="light">(</span>30<span class="light">)</span></span> |- |<span class="venn">[[File:Venn 0011 0100.svg|20px]]<br><span class="light">(</span>44<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1100.svg|20px]]<br><span class="light">(</span>56<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0010.svg|20px]]<br><span class="light">(</span>74<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1010.svg|20px]]<br><span class="light">(</span>88<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0110.svg|20px]]<br><span class="light">(</span>98<span class="light">)</span></span><span class="venn">[[File:Venn 0010 0110.svg|20px]]<br><span class="light">(</span>100<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 0111.svg|20px]]<br><span class="light">(</span>228<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1011.svg|20px]]<br><span class="light">(</span>216<span class="light">)</span></span><span class="venn">[[File:Venn 0100 0111.svg|20px]]<br><span class="light">(</span>226<span class="light">)</span></span><span class="venn">[[File:Venn 0001 1101.svg|20px]]<br><span class="light">(</span>184<span class="light">)</span></span><span class="venn">[[File:Venn 0101 0011.svg|20px]]<br><span class="light">(</span>202<span class="light">)</span></span><span class="venn">[[File:Venn 0011 0101.svg|20px]]<br><span class="light">(</span>172<span class="light">)</span></span> |<span class="venn">[[File:Venn 0011 1001.svg|20px]]<br><span class="light">(</span>156<span class="light">)</span></span><span class="venn">[[File:Venn 0010 1101.svg|20px]]<br><span class="light">(</span>180<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1001.svg|20px]]<br><span class="light">(</span>154<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1011.svg|20px]]<br><span class="light">(</span>210<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0101.svg|20px]]<br><span class="light">(</span>166<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0011.svg|20px]]<br><span class="light">(</span>198<span class="light">)</span></span> |<span class="venn">[[File:Venn 0010 1100.svg|20px]]<br><span class="light">(</span>52<span class="light">)</span></span><span class="venn">[[File:Venn 0011 1000.svg|20px]]<br><span class="light">(</span>28<span class="light">)</span></span><span class="venn">[[File:Venn 0100 1010.svg|20px]]<br><span class="light">(</span>82<span class="light">)</span></span><span class="venn">[[File:Venn 0101 1000.svg|20px]]<br><span class="light">(</span>26<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0010.svg|20px]]<br><span class="light">(</span>70<span class="light">)</span></span><span class="venn">[[File:Venn 0110 0100.svg|20px]]<br><span class="light">(</span>38<span class="light">)</span></span> |} 9a4cqa31royh9j6wrh1tf912ftl0rsk 2690921 2690920 2024-12-08T22:57:55Z Watchduck 137431 2690921 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|<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 104</span> <span class="senior">(king 26752)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 104.svg|200px]] |- |<span class="venn-container">[[File:Venn 0001 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |} ===size 3=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 6</span> <span class="senior">(king 1632)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |- |<span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 110</span> <span class="senior">(king 28384)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |- |<span class="venn-container">[[File:Venn 0111 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span> |} ===size 4=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 22</span> <span class="senior">(king 5760)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |- |<span class="venn-container">[[File:Venn 0110 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 42</span> <span class="senior">(king 10920)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |- |<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0101 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0100 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span> |} ===size 6=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 8</span> <span class="senior">(king 2176)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |- |<span class="venn-container">[[File:Venn 0001 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 14</span> <span class="senior">(king 3808)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |- |<span class="venn-container">[[File:Venn 0111 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |} ===size 12=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 2</span> <span class="senior">(king 680)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |- |<span class="venn-container">[[File:Venn 0100 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0101 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 26</span> <span class="senior">(king 6856)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |- |<span class="venn-container">[[File:Venn 0101 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0100 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0001 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 44</span> <span class="senior">(king 11464)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |- |<span class="venn-container">[[File:Venn 0111 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0111 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |- |<span class="venn-container">[[File:Venn 0011 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0011 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span> |<span class="venn-container">[[File:Venn 0010 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span> |} 6978r8n8cscvu8qmncuzovfr6sx4kj8 2690922 2690921 2024-12-08T23:08:20Z Watchduck 137431 2690922 wikitext text/x-wiki <templatestyles src="Collapsible with classes/style.css" /> <templatestyles src="Principalities and dominions/overview/style.css" /> ===size 1=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 0</span> <span class="senior">(king 0)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 0.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 104</span> <span class="senior">(king 26752)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 104.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |} ===size 3=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 6</span> <span class="senior">(king 1632)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 110</span> <span class="senior">(king 28384)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span> |} ===size 4=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 22</span> <span class="senior">(king 5760)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 42</span> <span class="senior">(king 10920)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span> |} ===size 6=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 8</span> <span class="senior">(king 2176)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 14</span> <span class="senior">(king 3808)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |} ===size 12=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 2</span> <span class="senior">(king 680)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0100 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0101 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 26</span> <span class="senior">(king 6856)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0101 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 44</span> <span class="senior">(king 11464)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|20px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|20px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|20px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|20px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|20px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1000.svg|20px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|20px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|20px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span> |} dp91t6upz2i6j15fy69qxoufif4y5uo 2690923 2690922 2024-12-08T23:10:39Z Watchduck 137431 2690923 wikitext text/x-wiki <templatestyles src="Collapsible with classes/style.css" /> <templatestyles src="Principalities and dominions/overview/style.css" /> ===size 1=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 0</span> <span class="senior">(king 0)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 0.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 104</span> <span class="senior">(king 26752)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 104.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |} ===size 3=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 6</span> <span class="senior">(king 1632)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 110</span> <span class="senior">(king 28384)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span> |} ===size 4=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 22</span> <span class="senior">(king 5760)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 42</span> <span class="senior">(king 10920)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span> |} ===size 6=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 8</span> <span class="senior">(king 2176)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 14</span> <span class="senior">(king 3808)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |} ===size 12=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 2</span> <span class="senior">(king 680)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0100 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0101 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 26</span> <span class="senior">(king 6856)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0101 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 44</span> <span class="senior">(king 11464)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span> |} 0nqv9zgfovenkfry1vfnjgm2w9wv9gg 2690925 2690923 2024-12-08T23:13:03Z Watchduck 137431 2690925 wikitext text/x-wiki <templatestyles src="Collapsible with classes/style.css" /> <templatestyles src="Principalities and dominions/overview/style.css" /> ===size 1=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 0</span> <span class="senior">(king 0)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 0.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 0.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>0<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 104</span> <span class="senior">(king 26752)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z principality; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3T dominion; faction size 1; king index 104.svg|200px]] |class="image"| [[File:3Z dominion; faction size 1; king index 104.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |} ===size 3=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 6</span> <span class="senior">(king 1632)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 6.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 6.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 110</span> <span class="senior">(king 28384)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z principality; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3T dominion; faction size 3; king index 110.svg|200px]] |class="image"| [[File:3Z dominion; faction size 3; king index 110.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span> |} ===size 4=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 22</span> <span class="senior">(king 5760)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 22.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 22.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0110 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>22<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>150<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>232<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>104<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 42</span> <span class="senior">(king 10920)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z principality; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3T dominion; faction size 4; king index 42.svg|200px]] |class="image"| [[File:3Z dominion; faction size 4; king index 42.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>126<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span> |} ===size 6=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 8</span> <span class="senior">(king 2176)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 8.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 8.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>40<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>72<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>96<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 14</span> <span class="senior">(king 3808)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z principality; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3T dominion; faction size 6; king index 14.svg|200px]] |class="image"| [[File:3Z dominion; faction size 6; king index 14.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>14<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>50<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>84<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>238<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>250<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>252<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>144<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>132<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>130<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>112<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>76<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>42<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>20<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>18<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>6<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>60<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>90<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>102<span class="light">)</span></span></span> |} ===size 12=== {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 2</span> <span class="senior">(king 680)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 2.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 2.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0100 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>2<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>4<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>16<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0101 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>170<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>204<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>240<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>192<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>160<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>136<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>64<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>32<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>8<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0000 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>80<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0000 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>48<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>68<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>12<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>34<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>10<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 26</span> <span class="senior">(king 6856)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 26.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 26.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0101 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0001 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>116<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>92<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>114<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>58<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>78<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>46<span class="light">)</span></span></span> |} {| class="collapsible-with-classes collapsible open wide center wikitable" |- !colspan="4"| <span class="junior">king index 44</span> <span class="senior">(king 11464)</span> |- class="light-heading" |colspan="2"| principality ||colspan="2"| dominion |- class="light-heading" | truth tables || Zhegalkin indices || truth tables || Zhegalkin indices |- |class="image"| [[File:3T principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z principality; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3T dominion; faction size 12; king index 44.svg|200px]] |class="image"| [[File:3Z dominion; faction size 12; king index 44.svg|200px]] |- |class="faction"|<span class="venn-container">[[File:Venn 0111 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>62<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>94<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>118<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0111 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>190<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>222<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>246<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>212<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>178<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>142<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>124<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>122<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>110<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0101 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>106<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>108<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>120<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0100 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>66<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>36<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>24<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0110 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>86<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>54<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0111 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>30<span class="light">)</span></span></span> |- |class="faction"|<span class="venn-container">[[File:Venn 0011 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>44<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>56<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>74<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>88<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>98<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 0110.svg|25px]]<br><span class="venn-label"><span class="light">(</span>100<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>228<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>216<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 0111.svg|25px]]<br><span class="venn-label"><span class="light">(</span>226<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0001 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>184<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>202<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>172<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0011 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>156<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0010 1101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>180<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1001.svg|25px]]<br><span class="venn-label"><span class="light">(</span>154<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>210<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0101.svg|25px]]<br><span class="venn-label"><span class="light">(</span>166<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0011.svg|25px]]<br><span class="venn-label"><span class="light">(</span>198<span class="light">)</span></span></span> |class="faction"|<span class="venn-container">[[File:Venn 0010 1100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>52<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0011 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>28<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0100 1010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>82<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0101 1000.svg|25px]]<br><span class="venn-label"><span class="light">(</span>26<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0010.svg|25px]]<br><span class="venn-label"><span class="light">(</span>70<span class="light">)</span></span></span><span class="venn-container">[[File:Venn 0110 0100.svg|25px]]<br><span class="venn-label"><span class="light">(</span>38<span class="light">)</span></span></span> |}<noinclude> [[Category:Principalities and dominions of Boolean functions]] </noinclude> 4a1yajuf166yaavd0jcjliak25itqr8 3 317086 2690931 2024-12-09T01:13:02Z MathXplore 2888076 New resource with "== December 2024 == {{subst:VandalismBlock}}" 2690931 wikitext text/x-wiki == December 2024 == <div style="clear: both"></div>[[Image:Stop hand nuvola.svg|left|30px| ]] '''You have been <u>[[Wikiversity:Blocking policy|blocked]]</u> from editing for vandalizing Wikiversity.''' Please note that page blanking, addition of random text or spam, deliberate misinformation, privacy violations, and other deliberate attempts to disrupt Wikiversity are considered [[Wikiversity:vandalism|vandalism]]. If you wish to make useful contributions, you may come back if the block is set for a limited period of time or by requesting an unblock. If you are unblocked, you are expected to abide by the [[Wikiversity:Policies|Policies of Wikiversity]]. - [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 01:13, 9 December 2024 (UTC) j13b495014etlnfv44rhnqjni4cxsi0 6 317087 2690941 2024-12-09T02:17:00Z Young1lim 21186 {{Information |Description=LIB.2A: Shared Libraries (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690941 wikitext text/x-wiki == Summary == {{Information |Description=LIB.2A: Shared Libraries (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} 2hypa9qih8aytf7wjnbw8mb0jl7vgtd 6 317088 2690949 2024-12-09T03:20:42Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241205 - 20241204) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690949 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241205 - 20241204) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} jfpyspeupaj0ubsj5qukd5jcw6f3bcb 6 317089 2690951 2024-12-09T03:21:42Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241206 - 20241205) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690951 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241206 - 20241205) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} twbo0juem240k3rqdqajax0r5cotref 6 317090 2690953 2024-12-09T03:22:35Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241207 - 20241206) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690953 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241207 - 20241206) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} 8msn9345h6j23oidkqhqpr6ez89ch81 6 317091 2690955 2024-12-09T03:23:22Z Young1lim 21186 {{Information |Description=LCal.9A: Recursion (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690955 wikitext text/x-wiki == Summary == {{Information |Description=LCal.9A: Recursion (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} 17o67dheb7ldrxt3b7wqxmfwtzikgbd 6 317097 2690967 2024-12-09T04:15:49Z Young1lim 21186 {{Information |Description=ARM.2ASM: Branch and Return Methods (20241209- 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690967 wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Branch and Return Methods (20241209- 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} 30wb73xmdhb7ivxb7wf7onrjsyae70k 6 317098 2690969 2024-12-09T04:34:30Z Young1lim 21186 {{Information |Description=Work2.1A: Libraries (20241209 - 20241109) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690969 wikitext text/x-wiki == Summary == {{Information |Description=Work2.1A: Libraries (20241209 - 20241109) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} iqgtfbashjb8bdgsll0lubajno3kt62 0 317099 2690973 2024-12-09T05:17:11Z Maha1devan 2982698 I added some basic material on verifying solutions to ODEs. 2690973 wikitext text/x-wiki Checking that a proposed function solves an ODE involves checking that the function satisfies the relation given by the ODE on the domain of interest, which is implicitly taken to be the whole real line in many cases. A function that solves an ODE is called a solution to the ODE. For example, consider the ODE <math display = "block">\frac{dy}{dx} = 0.</math> We propose that the following function solves the differential equation: <math>y(x) = 5</math>. Indeed, we find that <math display = "block"> \frac{d}{dx}(5) = 0,</math> so <math>y(x) = 5</math> solves this ODE. However, this is not the only function that solves this ODE. Examining the above calculation, the precise numerical value of the constant function was not used in the calculation, just the fact that the derivative of a constant function is zero. Therefore, any constant function would solve that ODE. Now consider another example, with the ODE being <math display = "block>\frac{d^2y}{dx^2} = -4y</math> We claim that the function <math>y(x) = \sin(2x)</math> solves this equation. We compute <math display = "block>\frac{d^2y}{dx^2} = \frac{d^2}{dx^2}\left(\sin (2x)\right) \frac{d}{dx}\left(\frac{d}{dx}\left(\sin (2x)\right)\right) = \frac{d}{dx}\left(2\cos (2x)\right) = -4\sin (2x) = -4y </math>, and note that this does satisfy the ODE, and is therefore a solution. Again, this is not the only solution to the ODE. For example, any constant multiple of <math>\sin(2x)</math> would also solve this ODE. Most of an ODEs course is devoted to the more difficult problem of finding an unknown function that satisfies a given ODE, and confirming that all possible solutions are known. Nonetheless, it is important to know what is meant by a function solving an ODE and how to check that a function solves an ODE before proceeding with this. h9emog6yvhnnzx9996qhytbh5e0xz6j 2690974 2690973 2024-12-09T05:21:54Z Maha1devan 2982698 fixed typo 2690974 wikitext text/x-wiki Checking that a proposed function solves an ODE involves checking that the function satisfies the relation given by the ODE on the domain of interest, which is implicitly taken to be the whole real line in many cases. A function that solves an ODE is called a solution to the ODE. For example, consider the ODE <math display = "block">\frac{dy}{dx} = 0.</math> We propose that the following function solves the differential equation: <math>y(x) = 5</math>. Indeed, we find that <math display = "block"> \frac{d}{dx}(5) = 0,</math> so <math>y(x) = 5</math> solves this ODE. However, this is not the only function that solves this ODE. Examining the above calculation, the precise numerical value of the constant function was not used in the calculation, just the fact that the derivative of a constant function is zero. Therefore, any constant function would solve that ODE. Now consider another example, with the ODE being <math display = "block>\frac{d^2y}{dx^2} = -4y</math> We claim that the function <math>y(x) = \sin(2x)</math> solves this equation. We compute <math display="block">\frac{d^2y}{dx^2} = \frac{d^2}{dx^2}\left(\sin (2x)\right) = \frac{d}{dx}\left(\frac{d}{dx}\left(\sin (2x)\right)\right) = \frac{d}{dx}\left(2\cos (2x)\right) = -4\sin (2x) = -4y </math>, and note that this does satisfy the ODE, and is therefore a solution. Again, this is not the only solution to the ODE. For example, any constant multiple of <math>\sin(2x)</math> would also solve this ODE. Most of an ODEs course is devoted to the more difficult problem of finding an unknown function that satisfies a given ODE, and confirming that all possible solutions are known. Nonetheless, it is important to know what is meant by a function solving an ODE and how to check that a function solves an ODE before proceeding with this. 72tan14ibrh0cj8s8msq3mngpv61a6s 0 317100 2690979 2024-12-09T06:40:43Z Maha1devan 2982698 Wrote up some material on IVPs 2690979 wikitext text/x-wiki In [[https://en.wikiversity.org/wiki/Ordinary_Differential_Equations/Verifying_Solutions_to_ODEs| the previous article]], we saw how to verify whether or not a function solved an ODE, and while doing so, found that many differential equations have infinitely many solutions. This means that as it stands, it is not meaningful to speak of "the" solution to an ODE, which is a big restriction on their utility. This suggests that additional information needs to be specified in order to distinguish between solutions of an ODE and select a unique one for a particular situation. The most common additional data that are specified are the initial conditions. An initial condition is an equation that specifies the value of the solution, and possibly its derivatives, at some point. In the context of problems where the dependent variable is time, an initial condition specifies the value of the solution and its derivatives at 0, or in other words, specifies the initial value of the solution, which motivates the following: An Initial Value Problem (IVP) is a differential equation combined with initial conditions. Here is an example of an IVP: Find a function <math>y(x)</math> such that <math display = "block">\begin{cases} \frac{dy}{dx} = y \\ y(0) = 1. \end{cases} </math> Later in the course, it will be justified that these are the only solutions, but we can verify that functions of the form <math>y(x) = Ce^x</math> for some real constant <math>C</math> solve this differential equation. We compute <math display = "block">\frac{dy}{dx} = \frac{d}{dx}(Ce^x) = C\frac{d}{dx}(e^x) = Ce^x = y,</math> so this solves the ODE. Now, we need to find which of these solutions also satisfy the initial condition. To do this, we assume <math>y</math> satisfies the initial conditions and find out for which value of <math>C</math> this is the case. If <math>y</math> satisfies the initial condition, then <math>y(0) = 1</math>, so <math display = "block>y(0) = Ce^0 = C = 1.</math> Then, the solution to the IVP is <math display = "block">y(x) = e^x.</math> The number of initial conditions needed to specify a unique solution depends on the order of the ODE. In general, <math>n</math> initial conditions need to be specified to create an IVP for an <math>n^{\text{th}}</math> order ODE with a unique solution. 8n6pgmbgduqpdppw793w09qv1zrostr 2690984 2690979 2024-12-09T07:26:04Z Maha1devan 2982698 Rewrite for clarity 2690984 wikitext text/x-wiki In [[https://en.wikiversity.org/wiki/Ordinary_Differential_Equations/Verifying_Solutions_to_ODEs| the previous article]], we saw how to verify whether or not a function solved an ODE, and while doing so, found that many differential equations have infinitely many solutions. This means that as it stands, it is not meaningful to speak of "the" solution to an ODE, which is a big restriction on their utility. This suggests that additional information needs to be specified in order to distinguish between solutions of an ODE and select a unique one for a particular situation. The most common additional data that are specified are the initial conditions. An initial condition is an equation that specifies the value of the solution, and possibly its derivatives, at some point. In the context of problems where the dependent variable is time, an initial condition specifies the value of the solution and its derivatives at 0, or in other words, specifies the initial value of the solution, which motivates the following: An Initial Value Problem (IVP) is a differential equation combined with initial conditions. Here is an example of an IVP: Find a function <math>y(x)</math> such that <math display = "block">\begin{cases} \frac{dy}{dx} = y \\ y(0) = 1. \end{cases} </math> The first line is the ODE, and the second line is the initial condition. To begin solving this IVP, we will begin by solving the ODE. Later in the course, it will be justified that our proposed solutions are the only solutions, but we can verify that functions of the form <math>y(x) = Ce^x</math> for some real constant <math>C</math> solve this differential equation. We compute <math display = "block">\frac{dy}{dx} = \frac{d}{dx}(Ce^x) = C\frac{d}{dx}(e^x) = Ce^x = y,</math> so this solves the ODE. Now, we need to find which of these solutions also satisfy the initial condition. To do this, we assume <math>y</math> satisfies the initial conditions and find out for which value of <math>C</math> this is the case. If <math>y</math> satisfies the initial condition, then <math>y(0) = 1</math>, so <math display = "block>y(0) = Ce^0 = C = 1.</math> Then, the solution to the IVP is <math display = "block">y(x) = e^x.</math> The number of initial conditions needed to specify a unique solution depends on the order of the ODE. In general, <math>n</math> initial conditions need to be specified to create an IVP for an <math>n^{\text{th}}</math> order ODE with a unique solution. ggjmhkhoj2cz6mmombn6cvbwvw1y5qp 6 317102 2690992 2024-12-09T09:04:09Z Young1lim 21186 {{Information |Description=Borrows (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |Author=Young W. Lim |Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} }} 2690992 wikitext text/x-wiki == Summary == {{Information |Description=Borrows (20241209 - 20241207) |Source={{own|Young1lim}} |Date=2024-12-09 |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}} fjyda60u7xqb7cqngka3cw5spw7r1td